# Evaluate the limit.

## Homework Statement

Evaluate the lim as x approaches 3 of (x/x-3) times the integral from 3 to x of (sint/t)dt

## The Attempt at a Solution

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So, what did you already do to solve the problem?
How do you usually solve "0/0" situations?

I tried to evaluate this using the substitution method: making u=sint and du=costdt. Then that created a big jumbled incorrect mess. I don't know how to attack this problem.

One of the problems with this particular integral, is that you cannot solve it. So don't bother with solving this integral, it won't work. But there is a way to make this integral disappear: differentiate it. Now, I wonder, is there a way that you can solve a limit using derivatives...

[f(x+h)-f(x)]/h ? I'm really not sure. What would be the h? 3? I'm so lost.

How would you resolve a "0/0"-situation with the aid of derivatives????

L'hopital rule

L'hospital rule: so take the derivative of the top and the bottom of (x/x-3), thus giving you 1/1 times the integral. How will this make the integral disappear?

$$\lim_{x\rightarrow 3}{\frac{x\int_3^x{\frac{\sin(t)}{t}dt}}{x-3}}$$

Try applying l'Hopitals rule on that. You are correct that the integral will not disappear, but the limit will become simpler...

Ok, so after applying L'Hopitals rule, I ended up with the limit as x approaches 3 of [sin(x)/x + sinx]. That can't be right. I'm sorry that I am mathematically incompetent.

no, that can't be right, there should still be an integral in there...
For ease, define

$$F(x)=\int_3^x{\frac{\sin(t)}{t}dt}$$

What is $$F^\prime(x)$$ (this is basically the fundamental theorem of calculus).

Now you want to calculate the derivative of xF(x). How would you do this? (hint: product rule)

According to the fundamental theorem of calculus, F'(x) equals f(x) or sin(x)/x. Now, upon calculating the derivative of xF(x), I use the product rule which will be x'F(x) + xF'(x). That gives me the integral from 3 to x of sin(t)/t dt + sinx.

Yes, that is correct. Now take the limit of that expression. What do you get?

Well, x is approaching 3, if I plug in 3, the integral will be from 3 to 3 making it zero. Then I'm left with sin(3)...

Yes, sin(3) seems to be the correct answer.

I cannot thank you enough for your patience and kindness. I truly appreciate your help. Thank you and God bless!