83:@0.942772:0.974518:0.963274:0.974518:0.963274:0.956333:0.942772:0.956333:0.000000:0.000000
Pensamiento numérico y variacional:@0.603717:0.043661:0.926913:0.043661:0.926913:0.024916:0.603717:0.024916:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
0:@0.492927:0.357492:0.501721:0.357492:0.501721:0.342109:0.492927:0.342109:0.000000
5:@0.601630:0.357492:0.610424:0.357492:0.610424:0.342109:0.601630:0.342109:0.000000
50:@0.484133:0.292322:0.501721:0.292322:0.501721:0.276939:0.484133:0.276939:0.000000:0.000000
100:@0.475339:0.234931:0.501721:0.234931:0.501721:0.219548:0.475339:0.219548:0.000000:0.000000:0.000000
150:@0.475339:0.177567:0.501721:0.177567:0.501721:0.162184:0.475339:0.162184:0.000000:0.000000:0.000000
-5:@0.396587:0.357479:0.410751:0.357479:0.410751:0.342096:0.396587:0.342096:0.000000:0.000000
(3,27):@0.582177:0.322559:0.635898:0.322559:0.635898:0.301407:0.582177:0.301407:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Figura 2.3:@0.463040:0.400389:0.536993:0.400389:0.536993:0.383719:0.463040:0.383719:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
2.4 Cálculo de límites con factorización:@0.073083:0.434893:0.377693:0.434893:0.377693:0.417946:0.073083:0.417946:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Al igual que en el ejemplo anterior, no todos los límites se pueden calcular por sustitución directa. Normalmente, el :@0.073083:0.462255:0.930539:0.462255:0.930539:0.446542:0.073083:0.446542:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
cálculo de los límites requiere un proceso más elaborado para evaluarlos. En los ejemplos dados a continuación :@0.073083:0.478897:0.930476:0.478897:0.930476:0.463184:0.073083:0.463184:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
observamos cómo la factorización nos ayuda para acabar con una indeterminación después de cancelar el factor que :@0.073083:0.495538:0.931606:0.495538:0.931606:0.479826:0.073083:0.479826:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
generaba dicha indeterminación.:@0.073083:0.512180:0.314039:0.512180:0.314039:0.496468:0.073083:0.496468:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Ejemplo 6. :@0.073083:0.543120:0.156431:0.543120:0.156431:0.526451:0.073083:0.526451:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Calcular los siguientes límites::@0.156302:0.543120:0.370761:0.543120:0.370761:0.527407:0.156302:0.527407:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
a.  lim:@0.073089:0.576242:0.128177:0.573810:0.128177:0.558098:0.073089:0.560530:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x:@0.103290:0.580393:0.108712:0.580393:0.108712:0.568461:0.103290:0.568461:0.000000
2:@0.119462:0.580393:0.125995:0.580393:0.125995:0.568965:0.119462:0.568965:0.000000
x:@0.141385:0.570224:0.148840:0.570224:0.148840:0.553818:0.141385:0.553818:0.000000
–:@0.149871:0.570806:0.159075:0.570806:0.159075:0.553152:0.149871:0.553152:0.000000
2:@0.160087:0.570224:0.169070:0.570224:0.169070:0.554511:0.160087:0.554511:0.000000
x:@0.138459:0.587559:0.145913:0.587559:0.145913:0.571153:0.138459:0.571153:0.000000
2:@0.146961:0.581496:0.152198:0.581496:0.152198:0.572336:0.146961:0.572336:0.000000
–:@0.152796:0.588143:0.162000:0.588143:0.162000:0.570488:0.152796:0.570488:0.000000
4:@0.163012:0.587560:0.171995:0.587560:0.171995:0.571847:0.163012:0.571847:0.000000
        lim:@0.073093:0.620081:0.125629:0.617649:0.125629:0.601936:0.073093:0.604368:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x:@0.100743:0.624231:0.106165:0.624231:0.106165:0.612300:0.100743:0.612300:0.000000
2:@0.116915:0.624231:0.123447:0.624231:0.123447:0.612804:0.116915:0.612804:0.000000
x:@0.138838:0.614062:0.146293:0.614062:0.146293:0.597656:0.138838:0.597656:0.000000
–:@0.147324:0.614645:0.156528:0.614645:0.156528:0.596991:0.147324:0.596991:0.000000
2:@0.157540:0.614062:0.166523:0.614062:0.166523:0.598350:0.157540:0.598350:0.000000
x:@0.135912:0.631397:0.143367:0.631397:0.143367:0.614991:0.135912:0.614991:0.000000
2:@0.144414:0.625335:0.149651:0.625335:0.149651:0.616174:0.144414:0.616174:0.000000
–:@0.150248:0.631981:0.159451:0.631981:0.159451:0.614327:0.150248:0.614327:0.000000
4:@0.160464:0.631399:0.169446:0.631399:0.169446:0.615686:0.160464:0.615686:0.000000
=:@0.176478:0.620665:0.191811:0.620665:0.191811:0.603011:0.176478:0.603011:0.000000
x:@0.191717:0.624231:0.197139:0.624231:0.197139:0.612300:0.191717:0.612300:0.000000
2:@0.207891:0.624231:0.214424:0.624231:0.214424:0.612804:0.207891:0.612804:0.000000
lim:@0.191717:0.617649:0.216604:0.617649:0.216604:0.601936:0.191717:0.601936:0.000000:0.000000:0.000000
x:@0.255902:0.614062:0.263357:0.614062:0.263357:0.597656:0.255902:0.597656:0.000000
–:@0.264388:0.614645:0.273591:0.614645:0.273591:0.596991:0.264388:0.596991:0.000000
2:@0.274604:0.614062:0.283586:0.614062:0.283586:0.598350:0.274604:0.598350:0.000000
(:@0.222891:0.631613:0.228463:0.631613:0.228463:0.615564:0.222891:0.615564:0.000000
x:@0.229384:0.631399:0.236839:0.631399:0.236839:0.614993:0.229384:0.614993:0.000000
–:@0.237869:0.631981:0.247073:0.631981:0.247073:0.614327:0.237869:0.614327:0.000000
2:@0.248085:0.631399:0.257068:0.631399:0.257068:0.615686:0.248085:0.615686:0.000000
) (:@0.258099:0.631613:0.274348:0.631613:0.274348:0.615564:0.258099:0.615564:0.000000:0.000000:0.000000
x:@0.276188:0.631399:0.283643:0.631399:0.283643:0.614993:0.276188:0.614993:0.000000
+:@0.284674:0.631981:0.300007:0.631981:0.300007:0.614327:0.284674:0.614327:0.000000
2:@0.301020:0.631399:0.310002:0.631399:0.310002:0.615686:0.301020:0.615686:0.000000
):@0.311033:0.631613:0.316606:0.631613:0.316606:0.615564:0.311033:0.615564:0.000000
=:@0.319430:0.620664:0.334764:0.620664:0.334764:0.603009:0.319430:0.603009:0.000000
x:@0.334671:0.627359:0.340093:0.627359:0.340093:0.615428:0.334671:0.615428:0.000000
2:@0.350843:0.627359:0.357376:0.627359:0.357376:0.615932:0.350843:0.615932:0.000000
lim:@0.334671:0.620778:0.359558:0.620778:0.359558:0.605066:0.334671:0.605066:0.000000:0.000000:0.000000
1:@0.379749:0.613988:0.388732:0.613988:0.388732:0.598275:0.379749:0.598275:0.000000
x:@0.367324:0.631323:0.374779:0.631323:0.374779:0.614917:0.367324:0.614917:0.000000
+:@0.375810:0.631906:0.391143:0.631906:0.391143:0.614251:0.375810:0.614251:0.000000
2:@0.392156:0.631323:0.401138:0.631323:0.401138:0.615610:0.392156:0.615610:0.000000
=:@0.405519:0.620658:0.420853:0.620658:0.420853:0.603004:0.405519:0.603004:0.000000
1:@0.428133:0.614313:0.437116:0.614313:0.437116:0.598600:0.428133:0.598600:0.000000
4:@0.428133:0.630081:0.437116:0.630081:0.437116:0.614369:0.428133:0.614369:0.000000
b.  lim:@0.073089:0.663920:0.128177:0.661488:0.128177:0.645775:0.073089:0.648207:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x:@0.103290:0.668070:0.108712:0.668070:0.108712:0.656138:0.103290:0.656138:0.000000
5:@0.119462:0.668070:0.125995:0.668070:0.125995:0.656643:0.119462:0.656643:0.000000
x:@0.137948:0.657901:0.145403:0.657901:0.145403:0.641495:0.137948:0.641495:0.000000
3:@0.146442:0.651837:0.151679:0.651837:0.151679:0.642676:0.146442:0.642676:0.000000
–:@0.152276:0.658483:0.161479:0.658483:0.161479:0.640829:0.152276:0.640829:0.000000
125:@0.162492:0.657901:0.191502:0.657901:0.191502:0.642188:0.162492:0.642188:0.000000:0.000000:0.000000
x:@0.142943:0.675236:0.150398:0.675236:0.150398:0.658830:0.142943:0.658830:0.000000
2:@0.151446:0.669173:0.156683:0.669173:0.156683:0.660013:0.151446:0.660013:0.000000
–:@0.157279:0.675820:0.166483:0.675820:0.166483:0.658166:0.157279:0.658166:0.000000
25:@0.167495:0.675237:0.186491:0.675237:0.186491:0.659525:0.167495:0.659525:0.000000:0.000000
        lim:@0.073084:0.707758:0.125630:0.705328:0.125630:0.689615:0.073084:0.692046:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x:@0.100743:0.711909:0.106165:0.711909:0.106165:0.699977:0.100743:0.699977:0.000000
5:@0.116915:0.711909:0.123448:0.711909:0.123448:0.700481:0.116915:0.700481:0.000000
x:@0.135401:0.701741:0.142856:0.701741:0.142856:0.685335:0.135401:0.685335:0.000000
3:@0.143894:0.695677:0.149131:0.695677:0.149131:0.686516:0.143894:0.686516:0.000000
–:@0.149729:0.702323:0.158932:0.702323:0.158932:0.684669:0.149729:0.684669:0.000000
125:@0.159945:0.701741:0.188955:0.701741:0.188955:0.686028:0.159945:0.686028:0.000000:0.000000:0.000000
x:@0.140396:0.719076:0.147851:0.719076:0.147851:0.702670:0.140396:0.702670:0.000000
2:@0.148897:0.713012:0.154134:0.713012:0.154134:0.703852:0.148897:0.703852:0.000000
–:@0.154732:0.719658:0.163936:0.719658:0.163936:0.702004:0.154732:0.702004:0.000000
25:@0.164948:0.719076:0.183945:0.719076:0.183945:0.703363:0.164948:0.703363:0.000000:0.000000
=:@0.195210:0.708342:0.210543:0.708342:0.210543:0.690688:0.195210:0.690688:0.000000
x:@0.210454:0.711909:0.215876:0.711909:0.215876:0.699977:0.210454:0.699977:0.000000
5:@0.226626:0.711909:0.233159:0.711909:0.233159:0.700481:0.226626:0.700481:0.000000
lim:@0.210454:0.705328:0.235341:0.705328:0.235341:0.689615:0.210454:0.689615:0.000000:0.000000:0.000000
(:@0.242553:0.701955:0.248126:0.701955:0.248126:0.685906:0.242553:0.685906:0.000000
x:@0.249046:0.701741:0.256501:0.701741:0.256501:0.685335:0.249046:0.685335:0.000000
–:@0.257532:0.702323:0.266735:0.702323:0.266735:0.684669:0.257532:0.684669:0.000000
5:@0.267748:0.701741:0.276730:0.701741:0.276730:0.686028:0.267748:0.686028:0.000000
)(:@0.277761:0.701955:0.289826:0.701955:0.289826:0.685906:0.277761:0.685906:0.000000:0.000000
x:@0.290747:0.701741:0.298202:0.701741:0.298202:0.685335:0.290747:0.685335:0.000000
2:@0.299241:0.695677:0.304478:0.695677:0.304478:0.686516:0.299241:0.686516:0.000000
+:@0.305074:0.702323:0.320408:0.702323:0.320408:0.684669:0.305074:0.684669:0.000000
5:@0.321420:0.701741:0.330403:0.701741:0.330403:0.686028:0.321420:0.686028:0.000000
x:@0.331433:0.701741:0.338888:0.701741:0.338888:0.685335:0.331433:0.685335:0.000000
+:@0.339919:0.702323:0.355252:0.702323:0.355252:0.684669:0.339919:0.684669:0.000000
25:@0.356265:0.701741:0.375261:0.701741:0.375261:0.686028:0.356265:0.686028:0.000000:0.000000
):@0.376279:0.701955:0.381851:0.701955:0.381851:0.685906:0.376279:0.685906:0.000000
(:@0.267893:0.719290:0.273466:0.719290:0.273466:0.703241:0.267893:0.703241:0.000000
x:@0.274389:0.719076:0.281844:0.719076:0.281844:0.702670:0.274389:0.702670:0.000000
–:@0.282875:0.719658:0.292079:0.719658:0.292079:0.702004:0.282875:0.702004:0.000000
5:@0.293091:0.719076:0.302074:0.719076:0.302074:0.703363:0.293091:0.703363:0.000000
)(:@0.303105:0.719290:0.315170:0.719290:0.315170:0.703241:0.303105:0.703241:0.000000:0.000000
x:@0.316090:0.719076:0.323545:0.719076:0.323545:0.702670:0.316090:0.702670:0.000000
+:@0.324576:0.719658:0.339909:0.719658:0.339909:0.702004:0.324576:0.702004:0.000000
5:@0.340922:0.719076:0.349904:0.719076:0.349904:0.703363:0.340922:0.703363:0.000000
):@0.350935:0.719290:0.356508:0.719290:0.356508:0.703241:0.350935:0.703241:0.000000
=:@0.385603:0.708341:0.400936:0.708341:0.400936:0.690687:0.385603:0.690687:0.000000
x:@0.400844:0.711909:0.406266:0.711909:0.406266:0.699977:0.400844:0.699977:0.000000
5:@0.417015:0.711909:0.423548:0.711909:0.423548:0.700481:0.417015:0.700481:0.000000
lim:@0.400844:0.705328:0.425730:0.705328:0.425730:0.689615:0.400844:0.689615:0.000000:0.000000:0.000000
x:@0.434248:0.701741:0.441703:0.701741:0.441703:0.685335:0.434248:0.685335:0.000000
2:@0.442740:0.695677:0.447977:0.695677:0.447977:0.686516:0.442740:0.686516:0.000000
+:@0.448576:0.702323:0.463909:0.702323:0.463909:0.684669:0.448576:0.684669:0.000000
5:@0.464921:0.701741:0.473904:0.701741:0.473904:0.686028:0.464921:0.686028:0.000000
x:@0.474935:0.701741:0.482390:0.701741:0.482390:0.685335:0.474935:0.685335:0.000000
+:@0.483421:0.702323:0.498754:0.702323:0.498754:0.684669:0.483421:0.684669:0.000000
25:@0.499766:0.701741:0.518762:0.701741:0.518762:0.686028:0.499766:0.686028:0.000000:0.000000
x:@0.459583:0.719076:0.467038:0.719076:0.467038:0.702670:0.459583:0.702670:0.000000
+:@0.468069:0.719658:0.483402:0.719658:0.483402:0.702004:0.468069:0.702004:0.000000
5:@0.484414:0.719076:0.493397:0.719076:0.493397:0.703363:0.484414:0.703363:0.000000
=:@0.523861:0.708342:0.539194:0.708342:0.539194:0.690688:0.523861:0.690688:0.000000
5:@0.546280:0.702123:0.555262:0.702123:0.555262:0.686410:0.546280:0.686410:0.000000
2:@0.556285:0.696059:0.561522:0.696059:0.561522:0.686898:0.556285:0.686898:0.000000
+:@0.562120:0.702705:0.577453:0.702705:0.577453:0.685051:0.562120:0.685051:0.000000
5 5:@0.578466:0.702123:0.602944:0.702123:0.602944:0.686410:0.578466:0.686410:0.000000:0.000000:0.000000
( ):@0.588472:0.702337:0.609539:0.702337:0.609539:0.686288:0.588472:0.686288:0.000000:0.000000:0.000000
+:@0.610459:0.702705:0.625792:0.702705:0.625792:0.685051:0.610459:0.685051:0.000000
25:@0.626805:0.702123:0.645801:0.702123:0.645801:0.686410:0.626805:0.686410:0.000000:0.000000
5 5:@0.578375:0.719458:0.613717:0.719458:0.613717:0.703745:0.578375:0.703745:0.000000:0.000000:0.000000
+:@0.588389:0.720040:0.603722:0.720040:0.603722:0.702386:0.588389:0.702386:0.000000
=:@0.653127:0.708336:0.668460:0.708336:0.668460:0.690682:0.653127:0.690682:0.000000
25  25 25:@0.676904:0.702123:0.773671:0.702123:0.773671:0.686410:0.676904:0.686410:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
+:@0.701975:0.702705:0.717308:0.702705:0.717308:0.685051:0.701975:0.685051:0.000000
+:@0.738329:0.702705:0.753662:0.702705:0.753662:0.685051:0.738329:0.685051:0.000000
10:@0.715780:0.716254:0.734777:0.716254:0.734777:0.700542:0.715780:0.700542:0.000000:0.000000
=:@0.782359:0.708336:0.797693:0.708336:0.797693:0.690682:0.782359:0.690682:0.000000
75:@0.802346:0.701990:0.821343:0.701990:0.821343:0.686278:0.802346:0.686278:0.000000:0.000000
10:@0.802346:0.717759:0.821343:0.717759:0.821343:0.702046:0.802346:0.702046:0.000000:0.000000
=:@0.825908:0.708342:0.841241:0.708342:0.841241:0.690688:0.825908:0.690688:0.000000
7:@0.841149:0.707760:0.850131:0.707760:0.850131:0.692047:0.841149:0.692047:0.000000
1:@0.857419:0.701990:0.866402:0.701990:0.866402:0.686278:0.857419:0.686278:0.000000
2:@0.857419:0.717759:0.866402:0.717759:0.866402:0.702046:0.857419:0.702046:0.000000
.:@0.874188:0.707760:0.877391:0.707760:0.877391:0.692047:0.874188:0.692047:0.000000
2.5 Cálculo de límites con racionalización:@0.073104:0.749420:0.394228:0.749420:0.394228:0.732473:0.073104:0.732473:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Ejemplo 7. :@0.073104:0.780360:0.156452:0.780360:0.156452:0.763690:0.073104:0.763690:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Utilicemos la racionalización para calcular los límites siguientes::@0.156324:0.780360:0.613578:0.780360:0.613578:0.764647:0.156324:0.764647:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
a.  lim:@0.073089:0.810815:0.128177:0.811040:0.128177:0.795327:0.073089:0.795103:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
t:@0.103291:0.817621:0.107200:0.817621:0.107200:0.805689:0.103291:0.805689:0.000000
1:@0.117950:0.817621:0.124483:0.817621:0.124483:0.806193:0.117950:0.806193:0.000000
√ –:@0.137460:0.808035:0.172489:0.808035:0.172489:0.790381:0.137460:0.790381:0.000000:0.000000:0.000000
t:@0.156879:0.807453:0.162254:0.807453:0.162254:0.791047:0.156879:0.791047:0.000000
1:@0.173501:0.807453:0.182484:0.807453:0.182484:0.791740:0.173501:0.791740:0.000000
t:@0.147160:0.824788:0.152535:0.824788:0.152535:0.808382:0.147160:0.808382:0.000000
–:@0.153566:0.825371:0.162769:0.825371:0.162769:0.807716:0.153566:0.807716:0.000000
1:@0.163782:0.824788:0.172765:0.824788:0.172765:0.809076:0.163782:0.809076:0.000000
En este caso, debemos racionalizar el numerador de la expresión racional y, luego, :@0.073089:0.841753:0.710608:0.841753:0.710608:0.826040:0.073089:0.826040:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
simplificar. Entonces::@0.073089:0.858395:0.223185:0.858395:0.223185:0.842682:0.073089:0.842682:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
t:@0.073089:0.898422:0.076998:0.898422:0.076998:0.886490:0.073089:0.886490:0.000000
1:@0.087748:0.898422:0.094281:0.898422:0.094281:0.886994:0.087748:0.886994:0.000000
lim:@0.073089:0.891840:0.097976:0.891840:0.097976:0.876127:0.073089:0.876127:0.000000:0.000000:0.000000
√ –:@0.107258:0.887204:0.142287:0.887204:0.142287:0.869550:0.107258:0.869550:0.000000:0.000000:0.000000
t:@0.126677:0.886621:0.132052:0.886621:0.132052:0.870215:0.126677:0.870215:0.000000
1:@0.143299:0.886621:0.152282:0.886621:0.152282:0.870909:0.143299:0.870909:0.000000
t:@0.116958:0.903957:0.122333:0.903957:0.122333:0.887551:0.116958:0.887551:0.000000
–:@0.123364:0.904539:0.132568:0.904539:0.132568:0.886885:0.123364:0.886885:0.000000
1:@0.133580:0.903957:0.142563:0.903957:0.142563:0.888244:0.133580:0.888244:0.000000
=:@0.158231:0.891948:0.173565:0.891948:0.173565:0.874294:0.158231:0.874294:0.000000
t:@0.173472:0.898621:0.177382:0.898621:0.177382:0.886689:0.173472:0.886689:0.000000
1:@0.188131:0.898621:0.194664:0.898621:0.194664:0.887193:0.188131:0.887193:0.000000
lim:@0.173472:0.892039:0.198359:0.892039:0.198359:0.876326:0.173472:0.876326:0.000000:0.000000:0.000000
(√ –:@0.207737:0.886076:0.249908:0.886076:0.249908:0.868421:0.207737:0.868421:0.000000:0.000000:0.000000:0.000000
t:@0.234298:0.885493:0.239673:0.885493:0.239673:0.869087:0.234298:0.869087:0.000000
1:@0.250920:0.885493:0.259903:0.885493:0.259903:0.869780:0.250920:0.869780:0.000000
) :@0.260934:0.885707:0.270606:0.885707:0.270606:0.869658:0.260934:0.869658:0.000000:0.000000
(√ +:@0.270522:0.886076:0.318823:0.886076:0.318823:0.868421:0.270522:0.868421:0.000000:0.000000:0.000000:0.000000
t:@0.297084:0.885493:0.302459:0.885493:0.302459:0.869087:0.297084:0.869087:0.000000
1:@0.319835:0.885493:0.328818:0.885493:0.328818:0.869780:0.319835:0.869780:0.000000
):@0.329849:0.885707:0.335421:0.885707:0.335421:0.869658:0.329849:0.869658:0.000000
( –:@0.216969:0.905041:0.239720:0.905041:0.239720:0.887387:0.216969:0.887387:0.000000:0.000000:0.000000
t:@0.224111:0.904459:0.229486:0.904459:0.229486:0.888053:0.224111:0.888053:0.000000
1  :@0.240733:0.904459:0.260266:0.904459:0.260266:0.888746:0.240733:0.888746:0.000000:0.000000:0.000000
):@0.250746:0.904673:0.256318:0.904673:0.256318:0.888624:0.250746:0.888624:0.000000
(√ +:@0.261297:0.905041:0.309597:0.905041:0.309597:0.887387:0.261297:0.887387:0.000000:0.000000:0.000000:0.000000
t:@0.287858:0.904459:0.293233:0.904459:0.293233:0.888053:0.287858:0.888053:0.000000
1:@0.310610:0.904459:0.319593:0.904459:0.319593:0.888746:0.310610:0.888746:0.000000
):@0.320617:0.904673:0.326189:0.904673:0.326189:0.888624:0.320617:0.888624:0.000000
=:@0.340428:0.891948:0.355761:0.891948:0.355761:0.874294:0.340428:0.874294:0.000000
t:@0.355669:0.898621:0.359578:0.898621:0.359578:0.886689:0.355669:0.886689:0.000000
1:@0.370328:0.898621:0.376861:0.898621:0.376861:0.887193:0.370328:0.887193:0.000000
lim:@0.355669:0.892039:0.380556:0.892039:0.380556:0.876326:0.355669:0.876326:0.000000:0.000000:0.000000
(√:@0.419034:0.886076:0.444583:0.886076:0.444583:0.868421:0.419034:0.868421:0.000000:0.000000
t:@0.445595:0.885493:0.450970:0.885493:0.450970:0.869087:0.445595:0.869087:0.000000
) –:@0.452008:0.885707:0.473357:0.885707:0.473357:0.869658:0.452008:0.869658:0.000000:0.000000:0.000000
2 :@0.457496:0.879429:0.465041:0.879429:0.465041:0.870269:0.457496:0.870269:0.000000:0.000000
1:@0.473273:0.885493:0.482256:0.885493:0.482256:0.869780:0.473273:0.869780:0.000000
2:@0.483280:0.879429:0.488517:0.879429:0.488517:0.870269:0.483280:0.870269:0.000000
( –:@0.399164:0.905041:0.421915:0.905041:0.421915:0.887387:0.399164:0.887387:0.000000:0.000000:0.000000
t:@0.406306:0.904459:0.411681:0.904459:0.411681:0.888053:0.406306:0.888053:0.000000
1  :@0.422928:0.904459:0.442463:0.904459:0.442463:0.888746:0.422928:0.888746:0.000000:0.000000:0.000000
):@0.432943:0.904673:0.438515:0.904673:0.438515:0.888624:0.432943:0.888624:0.000000
(√ +:@0.443493:0.905041:0.491794:0.905041:0.491794:0.887387:0.443493:0.887387:0.000000:0.000000:0.000000:0.000000
t:@0.470055:0.904459:0.475430:0.904459:0.475430:0.888053:0.470055:0.888053:0.000000
1:@0.492807:0.904459:0.501789:0.904459:0.501789:0.888746:0.492807:0.888746:0.000000
):@0.502813:0.904673:0.508386:0.904673:0.508386:0.888624:0.502813:0.888624:0.000000
=:@0.522625:0.891948:0.537958:0.891948:0.537958:0.874294:0.522625:0.874294:0.000000
t:@0.537866:0.898621:0.541775:0.898621:0.541775:0.886689:0.537866:0.886689:0.000000
1:@0.552525:0.898621:0.559058:0.898621:0.559058:0.887193:0.552525:0.887193:0.000000
lim:@0.537866:0.892039:0.562752:0.892039:0.562752:0.876326:0.537866:0.876326:0.000000:0.000000:0.000000
t:@0.624131:0.884256:0.629505:0.884256:0.629505:0.867850:0.624131:0.867850:0.000000
–:@0.630545:0.884471:0.638912:0.884471:0.638912:0.868421:0.630545:0.868421:0.000000
1:@0.638828:0.884256:0.647811:0.884256:0.647811:0.868544:0.638828:0.868544:0.000000
( –:@0.581360:0.903810:0.604112:0.903810:0.604112:0.886156:0.581360:0.886156:0.000000:0.000000:0.000000
t:@0.588502:0.903228:0.593877:0.903228:0.593877:0.886822:0.588502:0.886822:0.000000
1  :@0.605124:0.903228:0.624658:0.903222:0.624658:0.887509:0.605124:0.887515:0.000000:0.000000:0.000000
):@0.615138:0.903436:0.620710:0.903436:0.620710:0.887387:0.615138:0.887387:0.000000
(√ +:@0.625688:0.903805:0.673989:0.903805:0.673989:0.886150:0.625688:0.886150:0.000000:0.000000:0.000000:0.000000
t:@0.652250:0.903222:0.657625:0.903222:0.657625:0.886816:0.652250:0.886816:0.000000
1:@0.675002:0.903222:0.683984:0.903222:0.683984:0.887509:0.675002:0.887509:0.000000
):@0.685008:0.903436:0.690581:0.903436:0.690581:0.887387:0.685008:0.887387:0.000000
=:@0.073089:0.938035:0.088422:0.938035:0.088422:0.920381:0.073089:0.920381:0.000000
t:@0.088330:0.942369:0.092239:0.942369:0.092239:0.930437:0.088330:0.930437:0.000000
1:@0.102989:0.942369:0.109522:0.942369:0.109522:0.930941:0.102989:0.930941:0.000000
lim:@0.088330:0.935787:0.113217:0.935787:0.113217:0.920074:0.088330:0.920074:0.000000:0.000000:0.000000
1:@0.148740:0.926038:0.157723:0.926038:0.157723:0.910325:0.148740:0.910325:0.000000
(√ +:@0.120779:0.945592:0.169080:0.945592:0.169080:0.927938:0.120779:0.927938:0.000000:0.000000:0.000000:0.000000
t:@0.147341:0.945010:0.152716:0.945010:0.152716:0.928604:0.147341:0.928604:0.000000
1:@0.170092:0.945010:0.179075:0.945010:0.179075:0.929297:0.170092:0.929297:0.000000
):@0.180109:0.945218:0.185681:0.945218:0.185681:0.929168:0.180109:0.929168:0.000000
=:@0.189150:0.938035:0.204484:0.938035:0.204484:0.920381:0.189150:0.920381:0.000000
1:@0.230732:0.925984:0.239715:0.925984:0.239715:0.910271:0.230732:0.910271:0.000000
√ +:@0.207852:0.945538:0.252619:0.945538:0.252619:0.927883:0.207852:0.927883:0.000000:0.000000:0.000000
1 1:@0.227272:0.944955:0.262614:0.944955:0.262614:0.929243:0.227272:0.929243:0.000000:0.000000:0.000000
=:@0.265107:0.938035:0.280440:0.938035:0.280440:0.920381:0.265107:0.920381:0.000000
1:@0.287716:0.927035:0.296699:0.927035:0.296699:0.911322:0.287716:0.911322:0.000000
2:@0.287716:0.942803:0.296699:0.942803:0.296699:0.927091:0.287716:0.927091:0.000000
.:@0.304485:0.937464:0.307688:0.937464:0.307688:0.921751:0.304485:0.921751:0.000000
Límites de funciones :@0.741982:0.799393:0.877629:0.799393:0.877629:0.785527:0.741982:0.785527:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
indeterminadas:@0.741982:0.816035:0.844091:0.816035:0.844091:0.802169:0.741982:0.802169:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
t.ly/wCjCV:@0.765020:0.930570:0.839545:0.930570:0.839545:0.914857:0.765020:0.914857:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
MUESTRA EDITORIAL:@0.248902:0.947499:0.888171:0.113285:0.743922:0.050540:0.104653:0.884754:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
 Muestra editorial solo para fines didácticos – Prohibida su venta:@0.095585:0.659700:0.095585:0.341447:0.077558:0.341447:0.077558:0.659700:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000