Calculus Examples

$lim_{x \to \infty} \arctan (- e^{x})$

Step 1

Since the function $e^{x}$ approaches $\infty$ , the negative constant $- 1$ times the function approaches $- \infty$ .

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Consider the limit with the constant multiple $- 1$ removed.

$lim_{x \to \infty} e^{x}$

Since the exponent $x$ approaches $\infty$ , the quantity $e^{x}$ approaches $\infty$ .

$\infty$

Since the function $e^{x}$ approaches $\infty$ , the negative constant $- 1$ times the function approaches $- \infty$ .

$- \infty$

Step 2

Substitute $t$ for $- e^{x}$ and let $t$ approach $- \infty$ since $lim_{x \to \infty} - e^{x} = - \infty$ .

$lim_{t \to - \infty} \arctan (t)$

Step 3

The limit as $t$ approaches $- \infty$ is $- \frac{π}{2}$ .

$- \frac{π}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$- \frac{π}{2}$

Decimal Form:

$- 1.57079632 \dots$