Calculus Examples

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

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

Step 1

Since the function

e^{x}

$e^{x}$ approaches

\infty

$\infty$ , the negative constant

- 1

$- 1$ times the function approaches

- \infty

$- \infty$ .

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Step 1.1

Consider the limit with the constant multiple

- 1

$- 1$ removed.

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

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

Step 1.2

Since the exponent

x

$x$ approaches

\infty

$\infty$ , the quantity

e^{x}

$e^{x}$ approaches

\infty

$\infty$ .

\infty

$\infty$

Step 1.3

Since the function

e^{x}

$e^{x}$ approaches

\infty

$\infty$ , the negative constant

- 1

$- 1$ times the function approaches

- \infty

$- \infty$ .

- \infty

$- \infty$

- \infty

$- \infty$

Step 2

Substitute

t

$t$ for

- e^{x}

$- e^{x}$ and let

t

$t$ approach

- \infty

$- \infty$ since

lim_{x \to \infty} - e^{x} = - \infty

$lim_{x \to \infty} - e^{x} = - \infty$ .

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

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

Step 3

The limit as

t

$t$ approaches

- \infty

$- \infty$ is

- \frac{π}{2}

$- \frac{π}{2}$ .

- \frac{π}{2}

$- \frac{π}{2}$

[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$