# I'm wrong, but I can't figure out how.

1. May 9, 2009

### mr_garlic

1. The problem statement, all variables and given/known data
$$\lim_{x\rightarrow\infty} \sqrt{e^{2x}+9}-e^x$$

2. Relevant equations
$$\lim_{x\rightarrow\infty} \sqrt{x} = \infty$$
$$\lim_{x\rightarrow\infty} e^x = \infty$$

3. The attempt at a solution

$$\lim_{x\rightarrow\infty} \sqrt{e^{2x}+9}-e^x = \lim_{x\rightarrow\infty} \frac{e^{2x}+9-e^{2x}}{\sqrt{e^{2x}+9}+e^x}$$
$$=\lim_{x\rightarrow\infty} \frac{9}{\sqrt{e^{2x}+9}+e^x}$$

The limit as x>infty of sqrt(x) is infty
the limit as x>infty of e^2x is infty
the entire bottom term tends towards infinity as x tends towards infinity
therefore
$$\lim_{x\rightarrow\infty} \frac{9}{\sqrt{e^{2x}+9}+e^x}=0$$

My professor had a different answer that we didn't have time to go over in class, and my answer just doesn't feel right. Where did I make my mistake?

2. May 9, 2009

### Staff: Mentor

Looks fine to me. I don't see how your professor could come up with a different value unless the two of you started from different problems.

3. May 9, 2009

### mr_garlic

Thanks, I've been running through this again and again for the past couple days.