Calculus Examples

$lim_{x \to 8} \frac{\arctan (x)}{\frac{1}{x} - 4}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $8$ .

$\frac{lim_{x \to 8} \arctan (x)}{lim_{x \to 8} \frac{1}{x} - 4}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\frac{lim_{x \to 8} \arctan (x)}{lim_{x \to 8} \frac{1}{x} - lim_{x \to 8} 4}$

Step 3

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $8$ .

$\frac{lim_{x \to 8} \arctan (x)}{\frac{lim_{x \to 8} 1}{lim_{x \to 8} x} - lim_{x \to 8} 4}$

Step 4

Evaluate the limit of $1$ which is constant as $x$ approaches $8$ .

$\frac{lim_{x \to 8} \arctan (x)}{\frac{1}{lim_{x \to 8} x} - lim_{x \to 8} 4}$

Step 5

Evaluate the limit of $4$ which is constant as $x$ approaches $8$ .

$\frac{lim_{x \to 8} \arctan (x)}{\frac{1}{lim_{x \to 8} x} - 1 \cdot 4}$

Step 6

Evaluate the limits by plugging in $8$ for all occurrences of $x$ .

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Step 6.1

Evaluate the limit of $\arctan (x)$ by plugging in $8$ for $x$ .

$\frac{\arctan (8)}{\frac{1}{lim_{x \to 8} x} - 1 \cdot 4}$

Step 6.2

Evaluate $\arctan (8)$ .

$\frac{1.44644133}{\frac{1}{lim_{x \to 8} x} - 1 \cdot 4}$

Step 6.3

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{1.44644133}{\frac{1}{8} - 1 \cdot 4}$

Step 7

Simplify the answer.

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Step 7.1

Simplify the denominator.

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Step 7.1.1

Multiply $- 1$ by $4$ .

$\frac{1.44644133}{\frac{1}{8} - 4}$

Step 7.1.2

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$\frac{1.44644133}{\frac{1}{8} - 4 \cdot \frac{8}{8}}$

Step 7.1.3

Combine $- 4$ and $\frac{8}{8}$ .

$\frac{1.44644133}{\frac{1}{8} + \frac{- 4 \cdot 8}{8}}$

Step 7.1.4

Combine the numerators over the common denominator.

$\frac{1.44644133}{\frac{1 - 4 \cdot 8}{8}}$

Step 7.1.5

Simplify the numerator.

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Step 7.1.5.1

Multiply $- 4$ by $8$ .

$\frac{1.44644133}{\frac{1 - 32}{8}}$

Step 7.1.5.2

Subtract $32$ from $1$ .

$\frac{1.44644133}{\frac{- 31}{8}}$

Step 7.1.6

Move the negative in front of the fraction.

$\frac{1.44644133}{- \frac{31}{8}}$

Step 7.2

Multiply the numerator by the reciprocal of the denominator.

$1.44644133 (- \frac{8}{31})$

Step 7.3

Multiply $1.44644133 (- \frac{8}{31})$ .

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Step 7.3.1

Multiply $- 1$ by $1.44644133$ .

$- 1.44644133 (\frac{8}{31})$

Step 7.3.2

Combine $- 1.44644133$ and $\frac{8}{31}$ .

$\frac{- 1.44644133 \cdot 8}{31}$

Step 7.3.3

Multiply $- 1.44644133$ by $8$ .

$\frac{- 11.57153065}{31}$

Step 7.4

Divide $- 11.57153065$ by $31$ .

$- 0.37327518$