Limit of Sequence: Find n^2(e^\frac{1}{n^2} - cos(\frac{1}{n}))

[Dansuer]

Homework Statement

Find the limit of $n^2(e^\frac{1}{n^2} - cos(\frac{1}{n}))$

Homework Equations

The Attempt at a Solution

since cos(1/n) is asymptotic to 1. $n^2(e^\frac{1}{n^2} - cos(\frac{1}{n}))$ ~ $n^2(e^\frac{1}{n^2} - 1)$ ~ $n^2 \frac{1}{n^2})$ = 1
The right answer is 3/2 though. I don't see what's wrong with my reasoning. Maybe i used asymptotic in an illegitimate way. What's the problem?

 

[CompuChip]

You have to be a little more careful than that.
Try switching over to x = 1/n, then it will be the limit for x going to zero.
If you expand both terms inside the brackets in a series around 0, you can throw away terms of order x⁴ and you will arrive at the right answer.

 

[Dansuer]

Thanks, that way i solved it.
I also found what i did wrong with asymptotic. I though that when a sequence is asymptotic with another you could just substitute one with the other. But it's not true. in this case. $cos(1/n)$~ $1$ but $e^\frac{1}{n^2} - cos(\frac{1}{n})$ ~$\frac{3}{2} e^\frac{1}{n^2} - 1$.

$e^\frac{1}{n^2} - cos(\frac{1}{n})$ ~$e^\frac{1}{n^2} - 1$ This is not true.