Calculus Examples

$lim_{x \to 0} \frac{\arctan (x)}{(\frac{1}{x}) - 7}$

Step 1

Combine terms.

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Step 1.1

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$lim_{x \to 0} \frac{\arctan (x)}{\frac{1}{x} - 7 \cdot \frac{x}{x}}$

Step 1.2

Combine $- 7$ and $\frac{x}{x}$ .

$lim_{x \to 0} \frac{\arctan (x)}{\frac{1}{x} + \frac{- 7 x}{x}}$

Step 1.3

Combine the numerators over the common denominator.

$lim_{x \to 0} \frac{\arctan (x)}{\frac{1 - 7 x}{x}}$

Step 2

Simplify the limit argument.

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Step 2.1

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 0} \arctan (x) \frac{x}{1 - 7 x}$

Step 2.2

Combine $\arctan (x)$ and $\frac{x}{1 - 7 x}$ .

$lim_{x \to 0} \frac{\arctan (x) x}{1 - 7 x}$

Step 3

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $0$ .

$\frac{lim_{x \to 0} \arctan (x) x}{lim_{x \to 0} 1 - 7 x}$

Step 4

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{lim_{x \to 0} \arctan (x) \cdot lim_{x \to 0} x}{lim_{x \to 0} 1 - 7 x}$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{lim_{x \to 0} \arctan (x) \cdot lim_{x \to 0} x}{lim_{x \to 0} 1 - lim_{x \to 0} 7 x}$

Step 6

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$\frac{lim_{x \to 0} \arctan (x) \cdot lim_{x \to 0} x}{1 - lim_{x \to 0} 7 x}$

Step 7

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$\frac{lim_{x \to 0} \arctan (x) \cdot lim_{x \to 0} x}{1 - 7 lim_{x \to 0} x}$

Step 8

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 8.1

Evaluate the limit of $\arctan (x)$ by plugging in $0$ for $x$ .

$\frac{\arctan (0) \cdot lim_{x \to 0} x}{1 - 7 lim_{x \to 0} x}$

Step 8.2

The exact value of $\arctan (0)$ is $0$ .

$\frac{0 \cdot lim_{x \to 0} x}{1 - 7 lim_{x \to 0} x}$

Step 8.3

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{0 \cdot 0}{1 - 7 lim_{x \to 0} x}$

Step 8.4

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{0 \cdot 0}{1 - 7 \cdot 0}$

Step 9

Simplify the answer.

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Step 9.1

Multiply $0$ by $0$ .

$\frac{0}{1 - 7 \cdot 0}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

Multiply $- 7$ by $0$ .

$\frac{0}{1 + 0}$

Step 9.2.2

Add $1$ and $0$ .

$\frac{0}{1}$

Step 9.3

Divide $0$ by $1$ .

$0$