limn→+∞ e√n⋅(1−1√n)n
Answers are:
A)0
B)1
C)e
D)√e
E)1√e
I tried working on the second part to get it in a better form. In the end I got e−√n. Returning at the beginning with this new form it would be:
limn→+∞ e√n⋅e−√n which would eventually turn into limn→+∞ e0 so the answer would be B) 1 but it's not. The answer is E)1√e but I can't figure it out why.
Answer
Consider A=e√n∗(1−1√n)n
Take logarithms of both sides log(A)=√n+nlog(1−1√n)
Now, use the fact that for small values of x, log(1−x)=−x−x22+O(x3). Replace x by 1√n which makes log(A)=√n+n(−1√n−12n+⋯)
Expand and simplify.
I am sure that you can take from here.
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