# Thread: L'Hospital problem

1. ## L'Hospital problem

Another one:

this time im lost

2. Originally Posted by stiitches
Another one:

this time im lost
From your post title you are obviously aware of how it's to be done. So what have you tried and where are you stuck?

3. Recall, L'Hospital's rule states that if

$\displaystyle \displaystyle \lim_{x\to n}\frac{f(x)}{g(x)} = \frac{0}{0} \;\;\text{or}\;\; \frac{\pm \infty}{\pm \infty}\quad \text{then}\quad \lim_{x\to n} \frac{f(x)}{g(x)} \overset{\text{H}}{=}\lim_{x\to n}\frac{f'(x)}{g'(x)}$

So....

$\displaystyle \displaystyle \lim_{x\to 1}\frac{\ln(x)}{\sin(2\pi x)}\overset{\text{H}}{=}\lim_{x\to 1}\frac{\frac{d}{dx}\ln(x)}{\frac{d}{dx}\sin(2\pi x)}$

I'm confident you can handle it from here.

4. oops! i left our a few conditions.$\displaystyle f\;\text{and}\;g$ must be differentiable and $\displaystyle g'(x) \neq 0$.

But both are true here. So it shouldn't matter. Just bust out some sweet differentiation moves and you're golden.