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## Homework Statement

[tex]\lim_{x\rightarrow\infty} \sqrt{e^{2x}+9}-e^x[/tex]

## Homework Equations

[tex]\lim_{x\rightarrow\infty} \sqrt{x} = \infty[/tex]

[tex]\lim_{x\rightarrow\infty} e^x = \infty [/tex]

## The Attempt at a Solution

[tex]\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}[/tex]

[tex]=\lim_{x\rightarrow\infty} \frac{9}{\sqrt{e^{2x}+9}+e^x}[/tex]

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

[tex]\lim_{x\rightarrow\infty} \frac{9}{\sqrt{e^{2x}+9}+e^x}=0[/tex]

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?