limn→∞n(1−1e(1+1n)n)
If I write expansion of (1+1n)n it was equal to expansion of e so n−n=0. Is limit is zero ?
Edit:
Uploading screen shot of my response. They marked it correct and I don't know if answer key is wrong or not ? Please can anyone say for sure like with 100 percent surety if answer given is wrong.
Answer
By the L'Hôpital's rule we obtain:
limn→+∞n(1−1e(1+1n)n)=limx→0+e−(1+x)1xex=−1elimx→0+(eln(1+x)x)′=
=−1elimx→0+(1+x)1x(1x(1+x)−ln(1+x)x2)=
=−1elimx→0+(1+x)1xlimx→0+x−(1+x)ln(1+x)x2+x3=
=−limx→0+1−ln(1+x)−12x+3x2=limx→0+ln(1+x)xlimx→0+12+3x=12.
No comments:
Post a Comment