# Thread: [SOLVED] Limit and integral question

1. ## [SOLVED] Limit and integral question

I just had a killer calculus exam and there are two questions I cannot seem to figure out no matter what.

The first one goes:
The limit of (3+1/x)^x as x approaches 0+. Show what the limit is or if it does not exist show why.

I have no clue how to simplify it. I know it exists because it's just asking for the right sided limit, which means it must exist.

As for the integral question it goes like this:
find g'(x) if the g(x)= integral of pi/4 to sin(x^2) of arcsin(t)/(t^2+1) dx

I know that you cannot simply plug in pi/4 and sin(x^2) into the integral.. but I have no idea how to do it otherwise w/o antideriving it, plugging in the pi/4 and sin(x^2) and then deriving it again. But I had no clue how to antiderive arcsin(t).

Anyways, I keep mulling over these questions. If anyone could help me figure them out it would be great!!!
Thank you SO much!

2. Originally Posted by electricsparks
I just had a killer calculus exam and there are two questions I cannot seem to figure out no matter what.

The first one goes:
The limit of (3+1/x)^x as x approaches 0+. Show what the limit is or if it does not exist show why.

I have no clue how to simplify it. I know it exists because it's just asking for the right sided limit, which means it must exist.

As for the integral question it goes like this:
find g'(x) if the g(x)= integral of pi/4 to sin(x^2) of arcsin(t)/(t^2+1) dx

I know that you cannot simply plug in pi/4 and sin(x^2) into the integral.. but I have no idea how to do it otherwise w/o antideriving it, plugging in the pi/4 and sin(x^2) and then deriving it again. But I had no clue how to antiderive arcsin(t).

Anyways, I keep mulling over these questions. If anyone could help me figure them out it would be great!!!
Thank you SO much!

Are you asked to find $\displaystyle \int_{\frac{\pi}{4}}^{\sin{(x^2)}}{\frac{\arcsin{t }}{t^2 + 1}\,dt}$

and then take its derivative with respect to $\displaystyle x$?

3. Yes, that is what the second question is. Sorry, I didn't know how to make the integral sign on the computer. Thanks for helping me!

4. Originally Posted by electricsparks
I just had a killer calculus exam and there are two questions I cannot seem to figure out no matter what.

The first one goes:
The limit of (3+1/x)^x as x approaches 0+. Show what the limit is or if it does not exist show why.

I have no clue how to simplify it. I know it exists because it's just asking for the right sided limit, which means it must exist.

As for the integral question it goes like this:
find g'(x) if the g(x)= integral of pi/4 to sin(x^2) of arcsin(t)/(t^2+1) dx

I know that you cannot simply plug in pi/4 and sin(x^2) into the integral.. but I have no idea how to do it otherwise w/o antideriving it, plugging in the pi/4 and sin(x^2) and then deriving it again. But I had no clue how to antiderive arcsin(t).

Anyways, I keep mulling over these questions. If anyone could help me figure them out it would be great!!!
Thank you SO much!
1) Since $\displaystyle \left(3+ \frac{1}{x} \right)^x = e^{x \ln \left( 3 + \frac{1}{x}\right)}$ you should first consider $\displaystyle \lim_{x \to 0^+} x \ln \left( 3 + \frac{1}{x}\right) = \lim_{x \to 0^+} \frac{\ln \left( 3 + \frac{1}{x}\right)}{\frac{1}{x}}$ (I suggest using l'Hopital's Rule).

2) The technique for solving this type of question is shown in this thread: http://www.mathhelpforum.com/math-he...ion-parts.html