1. ## Limits

Here's the limit that I need to calculate :

lim x→y (x/x-y)*∫f(t)dt

where y is a real constant and f(t) is a continuous function. The integral is defined between y and x.

I haven't calculated limits for a while now, so I don't even know where to start. If someone could help me, that would be appreciated.

P.S. I don't need the answer, just a starting point

2. Originally Posted by yuki267
Here's the limit that I need to calculate :

lim x→y (x/x-y)*∫f(t)dt

where y is a real constant and f(t) is a continuous function. The integral is defined between y and x.

I haven't calculated limits for a while now, so I don't even know where to start. If someone could help me, that would be appreciated.

P.S. I don't need the answer, just a starting point
Is this what you are trying to say?

Find

$\displaystyle \lim_{x\to{y}}\left[\frac{x}{x-y}\cdot\int^x_yf(t)dt\right]$

where y is constant?

3. Yes.

4. Originally Posted by yuki267
Yes.
Well, that's a pickle.

It is clear that $\displaystyle \int_y^xf(t)dt\to0$ as $\displaystyle x\to{y}$.

And $\displaystyle \frac{x}{x-y}=\frac{1}{1-\frac{y}{x}}\to\infty$ as $\displaystyle x\to{y}$.

Which is strange...

This screams the second fundamental theorem though, but I don't know what to do. I'm gonna sit back and watch.

By the way, welcome to MHF.

5. Thanks a lot for your help. Hopefully someone will be able to figure it out.

6. See attachment for my best shot

7. I forgot to mention that i have to solve this problem without using L'Hopital. Sorry about that, but thanks anyways.

8. From the mean value theorem there exists a number $\displaystyle \alpha \in[x,y]$ such that $\displaystyle \int_{y}^{x}{f(t)dt}=f(\alpha)(x-y)$. So what happens with $\displaystyle \alpha$ when $\displaystyle x->y$ and use this to find the limit.

9. jetta bra -- tak sa mycket

forgive my spelling I lived in Sverige in 1976-77 and haven't had much practice

10. Frankly I think that we need to stipulate that $\displaystyle x>y$.
By the mean value theorem for integrals of continuous functions $\displaystyle \left( {\exists c_x \in \left[ {y,x} \right]} \right)\left[ {\frac{{\int_y^x {f(t)} }}{{x - y}} = f\left( {c_x } \right)} \right]$.
Now using this notation, the problem reduces to $\displaystyle \lim _{~x \to y^ + } xf\left( {c_x } \right)$.

11. Thank you all for helping me with this problem! I forgot all about the mean value theorem.

12. So, what - if anything - is wrong with Calculus26's approach? Is it not valid?

13. Originally Posted by VonNemo19
So, what - if anything - is wrong with Calculus26's approach? Is it not valid?
I do not like having to view an attachment.
I think that any serious responder should be able to avoid attachments by posting in LaTeX.
I guess that I just object to using attachments for anything other than graphs or very technical programming problems.

14. Originally Posted by Plato