Limits to infinity (with ln)

[WakelessFoil]

So I came across these two problems on my recent math quiz and I am just dead lost on them. I don't even really know what to call problems like these. I am familiar with limits and L'Hopital's rule (which was also on the quiz) but these two really got me scratching my head. Any help is appreciated!

$$\lim _{x\to \infty \:}\left(\left(1+\frac{1}{x^2}\right)^{2lnx}\right)$$
$$\lim _{x\to 0}\left(\left(x+e^{\frac{x}{15}}\right)^{\frac{15}{x}}\right)$$

 

[Dr.Peterson]

So I came across these two problems on my recent math quiz and I am just dead lost on them. I don't even really know what to call problems like these. I am familiar with limits and L'Hopital's rule (which was also on the quiz) but these two really got me scratching my head. Any help is appreciated!

$$\lim _{x\to \infty \:}\left(\left(1+\frac{1}{x^2}\right)^{2lnx}\right)$$
$$\lim _{x\to 0}\left(\left(x+e^{\frac{x}{15}}\right)^{\frac{15}{x}}\right)$$

For this type, you need to take a log before applying L'Hopital. See example 4 here,

Calculus I - L'Hospital's Rule and Indeterminate Forms

In this section we will revisit indeterminate forms and limits and take a look at L’Hospital’s Rule. L’Hospital’s Rule will allow us to evaluate some limits we were not able to previously.

tutorial.math.lamar.edu

and example 4 and problems 21-27 here:

L'Hopital's Rule

for two slightly different approaches.

Please show any work you can do, and tell us where you get stuck, so we can help you further.