Solve \lim _{x\to 0}\left(\frac{\int _0^x\:e^{t^2}dt}{\int _0^{2x}cos\left(\frac{t^2}{4}\right)dt\:}\right)

Question

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Answer 1

A problem of this sort is going to work well with the use of FTC part 1. Let's start by evaluating each of the integrals of the limit using FTC part 1:

For the numerator, the integral of e to the t² from bounds 0 to x will be evaluated by substituting x for wherever there is a t inside the integrand, and this term also needs to be multiplied by the derivative of the upper bound (in this case, it's 1).

Thus, the integral in the numerator is evaluated as e to the x².

Now we'll take a look at the denominator. We see our upper bound is 2x for this integral, so we will have to take this term and substitute it in place of the t in our integrand, cos(t²/4). Then we'll take that new term and multiply it by the derivative of the upper bound, in this case, 2.

Therefore, our denominator is evaluated as 2*cos(4x²/4), or, when simplified:

2*cos(x²)

Taking those evaluated integrals, we put them back into our limit. Always try to evaluate a limit first by simply plugging in whatever value that the limit is approaching. In this case, plug in 0.

When doing this, the numerator will become e⁰ which is equal to 1, and the denominator will become 2*cos(0) which is equal to 2*1 which is simply 2.

Therefore, our final limit as x approaches 0 evaluates to 1/2.

Hope this helped!