Need help with limits going to infinity Going into my Calculus final and there I'm having some issues with limits going to infinity and problems that are to the power of $x$.
Here is a problem I came across while studying which I'm pretty sure I messed up on.
Compute the following limit (a calculator answer is not sufficient):
$$\lim _{x\to\infty}(1+\sin(5/x))^x$$
Here's what I did:
First I set the formula to:
$e^{x\ln(1+\sin(5/x))}$
and figured then that if I find the limit for $x\ln(1+\sin(5/x))$ then I could solve the problem
So with that being said:
$f(x) = x\ln(1+\sin(5/x))
     = \ln(1+\sin(5/x))/(1/x)$ (honestly don't know how I got $1/x$, I just saw in a similar problem someone do that.
Then I got the derivative of it
so $f'(x) = (1/(1+\sin(5/x) (-5/x^2)\cos(5/x))/(-1/x^2)$
then I got rid of the $1/x^2$ by multiplying it by $x^2/1$, which would make $-1$ (if I'm correct?)
after multiplying the $-1$ into $-5/x^2$ I did the same thing with that, which then gave me
$5\cos(5/x)/(1+\sin(5/x)) = (5\cdot 1)/(1+0)= 5/1=5$
I then put $5$ into $e$ to make the final answer $e^5$.
Was this right? sorry if my formatting is bad, still am new to this. If i was wrong where did I mess up? This is one of the concepts im struggling with and am trying to nail it before my final hits.
Any help appreciated! :)
 A: We have $$\lim_{x\to\infty}\left(1+\sin\left(\frac5x\right)\right)^x.$$
The expression inside the limit (as you have done) can be written as $$\left(1+\sin\left(\frac5x\right)\right)^x=e^{x\log(1+\sin\left(5/x\right)}.$$
As such, we have $$\lim_{x\to\infty}e^{x\log(1+\sin\left(5/x\right)}= e^{\lim_{x\to\infty}x\log(1+\sin\left(5/x\right))}.$$
Focusing on the limit in the exponent, we have $$\lim_{x\to\infty}x\log(1+\sin(5/x))=\lim_{x\to\infty}\frac{\log(1+\sin(5/x))}{\frac1x}\overset{\text{L'H}}{=}\lim_{x\to\infty}\frac{\frac{1}{1+\sin(5/x)}\cdot\cos(5/x)\cdot-\frac{5}{x^2}}{-\frac{1}{x^2}}$$
$$=\lim_{x\to\infty}\frac{5\cos(5/x)}{1+\sin(5/x)}=5.$$
Therefore, we have $$\lim_{x\to\infty}\left(1+\sin\left(\frac5x\right)\right)^x=e^{\lim_{x\to\infty}x\log(1+\sin\left(5/x\right))}=e^5.$$
A: $$\lim_{x\to\infty}\left(1+\sin\left(\frac ax\right)\right)^x=\lim_{y\to 0}\left(1+\sin(y)\right)^{\frac a y}$$
So
$$A=\left(1+\sin(y)\right)^{\frac a y}\implies \log(A)={\frac a y}\log\left(1+\sin(y)\right)\sim a \frac {\sin(y)}y\sim a\implies A\sim e^a$$
