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Math Help - [SOLVED] Limit and integral question

  1. #1
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    [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!
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  2. #2
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    Quote Originally Posted by electricsparks View Post
    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!
    Your second question is hard to read.

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

    and then take its derivative with respect to x?
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  3. #3
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    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!
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  4. #4
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    Quote Originally Posted by electricsparks View Post
    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  \left(3+ \frac{1}{x} \right)^x = e^{x \ln \left( 3 + \frac{1}{x}\right)} you should first consider \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
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