Why Does My Calculation of the Limit Differ from My Professor's?

[mr_garlic]

Homework Statement

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

Homework Equations

$$\lim_{x\rightarrow\infty} \sqrt{x} = \infty$$
$$\lim_{x\rightarrow\infty} e^x = \infty$$

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?

 

[Mark44]

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.

 

[mr_garlic]

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