# Find the limit as x approaches infinity of y=arctan(e^x)?

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Jul 24, 2014

To answer this question, you need to know that ${\lim}_{x \to + \infty} {e}^{x} = + \infty$ and ${\lim}_{x \to + \infty} \arctan x = \frac{\pi}{2}$ from the stuy of ${e}^{x}$ (see Exponential functions ) and of $\arctan x$ (see inverse cosine and inverse tangent ).

So, as $x \to \infty$, ${e}^{x} \to \infty$ so that, letting $t = {e}^{x}$ we have
${\lim}_{x \to \infty} \arctan \left({e}^{x}\right) = {\lim}_{t \to \infty} \arctan \left(t\right) = \setminus \frac{\pi}{2}$.

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