Calculus Examples

$lim_{x \to \infty} \frac{\arctan (x)}{(\frac{1}{x}) - 2}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $\infty$ .

$\frac{lim_{x \to \infty} \arctan (x)}{lim_{x \to \infty} (\frac{1}{x}) - 2}$

Step 2

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

$\frac{\frac{π}{2}}{lim_{x \to \infty} (\frac{1}{x}) - 2}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\infty$ .

$\frac{\frac{π}{2}}{lim_{x \to \infty} \frac{1}{x} - lim_{x \to \infty} 2}$

Step 4

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x}$ approaches $0$ .

$\frac{\frac{π}{2}}{0 - lim_{x \to \infty} 2}$

Step 5

Evaluate the limit.

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Step 5.1

Evaluate the limit of $2$ which is constant as $x$ approaches $\infty$ .

$\frac{\frac{π}{2}}{0 - 1 \cdot 2}$

Step 5.2

Simplify the answer.

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Step 5.2.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{π}{2} \cdot \frac{1}{0 - 1 \cdot 2}$

Step 5.2.2

Simplify the denominator.

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Step 5.2.2.1

Multiply $- 1$ by $2$ .

$\frac{π}{2} \cdot \frac{1}{0 - 2}$

Step 5.2.2.2

Subtract $2$ from $0$ .

$\frac{π}{2} \cdot \frac{1}{- 2}$

Step 5.2.3

Move the negative in front of the fraction.

$\frac{π}{2} (- \frac{1}{2})$

Step 5.2.4

Multiply $\frac{π}{2} (- \frac{1}{2})$ .

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Step 5.2.4.1

Multiply $\frac{π}{2}$ by $\frac{1}{2}$ .

$- \frac{π}{2 \cdot 2}$

Step 5.2.4.2

Multiply $2$ by $2$ .

$- \frac{π}{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{π}{4}$

Decimal Form:

$- 0.78539816 \dots$