# Calculus 2 limits as x-->0+

#### JubJub1991

My teacher gave me this problem as an honors problem. She said that at one point I will have to use L'Hopitals rule to solve, but I have no idea where to start.

1. Evaluate the limit, if it exists.

l i m (x+1)^cotx
n-->0+

Any help would be greatly appreciated

#### [email protected]

MHF Hall of Honor
My teacher gave me this problem as an honors problem. She said that at one point I will have to use L'Hopitals rule to solve, but I have no idea where to start.

1. Evaluate the limit, if it exists.

l i m (x+1)^cotx
n-->0+

Any help would be greatly appreciated
Let $$\displaystyle f=x+1$$ and $$\displaystyle g=\cot x$$.

We end up with a limit of the form $$\displaystyle 1^\infty$$. To fix this notice $$\displaystyle \displaystyle f^g = \exp\left(\frac{\ln f}{1/g}\right)$$.

So $$\displaystyle \displaystyle \lim_{x\to0}f^g = \exp\left(\lim_{x\to0}\frac{\ln f}{1/g}\right)$$ and the limit is of the form $$\displaystyle \displaystyle \frac00$$ (can you see why?).

Now apply L'Hopital's rule.

• JubJub1991

#### JubJub1991

I cannot thank you enough! I did not see that it was f raised to the g. My only other question is on the second line you have f^g= exp(ln f/(1/g)). What is the exp part?

#### [email protected]

MHF Hall of Honor
$$\displaystyle \exp(x) = e^x$$

• JubJub1991