Evaluate the limit.

  • #1

Homework Statement



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

Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
22,089
3,293
So, what did you already do to solve the problem?
How do you usually solve "0/0" situations?
 
  • #3
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.
 
  • #4
22,089
3,293
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...
 
  • #5
[f(x+h)-f(x)]/h ? I'm really not sure. What would be the h? 3? I'm so lost.
 
  • #6
22,089
3,293
How would you resolve a "0/0"-situation with the aid of derivatives????

L'hopital rule
 
  • #7
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?
 
  • #8
22,089
3,293
Your limit is

[tex]\lim_{x\rightarrow 3}{\frac{x\int_3^x{\frac{\sin(t)}{t}dt}}{x-3}}[/tex]

Try applying l'Hopitals rule on that. You are correct that the integral will not disappear, but the limit will become simpler...
 
  • #9
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.
 
  • #10
22,089
3,293
no, that can't be right, there should still be an integral in there...
For ease, define

[tex]F(x)=\int_3^x{\frac{\sin(t)}{t}dt}[/tex]

What is [tex]F^\prime(x)[/tex] (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)
 
  • #11
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.
 
  • #12
22,089
3,293
Yes, that is correct. Now take the limit of that expression. What do you get?
 
  • #13
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)...
 
  • #14
22,089
3,293
Yes, sin(3) seems to be the correct answer.
 
  • #15
I cannot thank you enough for your patience and kindness. I truly appreciate your help. Thank you and God bless!
 

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