Limit of arctan and e

**darrenthe747** · February 5th 2009, 07:47 PM

Hello,

This is my first post on this forum

. I can't seem to decide what the limit of this relatively simple equation is:

lim x approaches minus infinity of [arctan(e^-x)]

Any help is greatly appreciated!

Darren

**Abu-Khalil** · February 5th 2009, 08:25 PM

Let $t=-x$ so when $x\to-\infty,t\to\infty$ and then $\lim_{x\to -\infty}\arctan\left(e^{-x}\right)=\lim_{t\to\infty}\arctan\left(e^t\right)$ , but $\arctan(x)$ and $e^x$ are continuos functions in $\mathbb{R}$ so we can rewrite the limit as $\arctan\left\{\exp\left(\lim_{t\to\infty}t\right)\ right\}=\arctan \left(\infty\right)=\frac{\pi}{2}$ .

**darrenthe747** · February 5th 2009, 09:02 PM

Thanks for the excellent response. It was a bit more involved than I thought. It makes perfect sense now.

Darren

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