Evaluating limits involving x and t [duplicate]

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Fundamental Theorem of Calculus for limx→0∫x0(x−t)sint2 dtxsin3x\lim\limits_{x\to 0}\frac{\int_0^x(x-t)\sin t^2\ dt}{x\sin^3x}

5 answers

Stuck while doing question 10, can’t proceed onto doing. Need some advice/solutions regarding this. Managed to solve question 11 already so don’t need solution for that question anymore.
Thanks in advance!

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1 Answer
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There are a couple of different ways to approach this:

1) When xx is small, the integration domain doesn’t go far, and you can develop the sine into a taylor series \sin t^2\approx t^2\sin t^2\approx t^2.

2) Use l’Hospital rule and differentiate both – just be careful, differentiation of the integral on top will have two terms: one with derivative under the integral sign and one differentiating over the upper limit. You may need to do this a couple of times, because both sides are of order 44.

3) You can try integrating by parts and see if you get anywhere.

  

 

I think that doing the integral is doable but impractical. I threw the integral into WolframAlpha and it did give me a result, but it was ugly. The Taylor Series does sound promising. How often can you assume things about a variable being integrated? I only do that for physical descriptions.
– O. Von Seckendorff
2 days ago

  

 

Taylor series is a natural way of solving limits, because in the limit they hold exactly. It’s usually the easiest way because Taylor series just means someone did the differentiation in advance so you don’t have to (it’s a lazy man’s version of l’Hospital). So here it should be the shortest way to the answer. The only thing you have to be careful about, is that you take enough terms to capture the lowest non-disappearing power exactly.
– orion
2 days ago

  

 

Interesting. I wasn’t taught that, but it does make a lot of sense.
– O. Von Seckendorff
2 days ago