Limit of x times integral of sin(t)/t from 3 to x

[PhysicsinCalifornia]

Integral in a limit??

Can anyone try to find the limit of this :

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

I know the answer is sin3. Can you show me the work involved (yes, i know the work also)

Hint: $$F'(3) = \lim_{x \rightarrow 3} \frac{F(x) - F(3)}{x-3}$$
*edited* F'(3)

 

[arildno]

So, since you know everything, what's your problem?
And, don't double post!

 

[dextercioby]

HINT:Use the theorem of Leibniz & Newton.

Daniel.

 

[Hurkyl]

I echo arildno -- what was the point of this post? Why should we bother telling you thinkgs you already know? It sounds an awfully lot like you're trying to get someone to do a homework problem for you.

 

[calculus1967]

HINT:Use the theorem of Leibniz & Newton.

Daniel.

Which one?

 

[saltydog]

What about L'Hospital's rule? And how about:

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

Negative infinity too. It looks like it approaches limits numerically but I don't know how to prove it. Suppose we could just drop the fractional term as that goes to one. Jesus, suppose I should just ask what is:

$$\int_a^{\infty} \frac{Sin(x)}{x}dx$$

Am I getting off the subject?

 

[PhysicsinCalifornia]

I echo arildno -- what was the point of this post? Why should we bother telling you thinkgs you already know? It sounds an awfully lot like you're trying to get someone to do a homework problem for you.

I'm sorry that I wasn't clear on my first post. Let me take this opportunity to correct myself.

I was solving problems and this one came up. I looked at the solution (because it has the solution also) but I didn't understand how to do it because I didn't learn from the book. I just needed help on the different approaches for this solution.

The hint that I added was the one in the book

Also, it is NOT a homework problem, but a problem for me to do just for fun( i guess)
The answer made no sense to me because it used that limit i put as a hint in the first post. I'm not sure if i see the connection

 

[shmoe]

The idea of the hint is to notice that the limit you're interested in is actually a limit like the kind in the definition of the derivative. Once you've identified what choice of F(x) will put it into this form, if you can differentiate F(x) in another way you'll be able to find this limit by simply evaluating this derivative at x=3.

In this case you already have the x-3 in the denominator, so you'll have to have

$$F(x)-F(3)=x\int_{3}^{x}\frac{\sin{t}}{t}dt$$

What choice of F(x) will work?