Calculus Examples

$\frac{lim_{x \to - 4} \frac{1}{4} + \frac{1}{x}}{4 + x}$

Step 1

Evaluate the limit.

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Step 1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 4$ .

$\frac{lim_{x \to - 4} \frac{1}{4} + lim_{x \to - 4} \frac{1}{x}}{4 + x}$

Step 1.2

Evaluate the limit of $\frac{1}{4}$ which is constant as $x$ approaches $- 4$ .

$\frac{\frac{1}{4} + lim_{x \to - 4} \frac{1}{x}}{4 + x}$

Step 1.3

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- 4$ .

$\frac{\frac{1}{4} + \frac{lim_{x \to - 4} 1}{lim_{x \to - 4} x}}{4 + x}$

Step 1.4

Evaluate the limit of $1$ which is constant as $x$ approaches $- 4$ .

$\frac{\frac{1}{4} + \frac{1}{lim_{x \to - 4} x}}{4 + x}$

Step 2

Evaluate the limit of $x$ by plugging in $- 4$ for $x$ .

$\frac{\frac{1}{4} + \frac{1}{- 4}}{4 + x}$

Step 3

Simplify the answer.

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Step 3.1

Multiply the numerator and denominator of the fraction by $4$ .

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Step 3.1.1

Multiply $\frac{\frac{1}{4} + \frac{1}{- 4}}{4 + x}$ by $\frac{4}{4}$ .

$\frac{4}{4} \cdot \frac{\frac{1}{4} + \frac{1}{- 4}}{4 + x}$

Step 3.1.2

Combine.

$\frac{4 (\frac{1}{4} + \frac{1}{- 4})}{4 (4 + x)}$

Step 3.2

Apply the distributive property.

$\frac{4 (\frac{1}{4}) + 4 (\frac{1}{- 4})}{4 \cdot 4 + 4 x}$

Step 3.3

Simplify by cancelling.

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Step 3.3.1

Cancel the common factor of $4$ .

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Step 3.3.1.1

Cancel the common factor.

$\frac{4 (\frac{1}{4}) + 4 (\frac{1}{- 4})}{4 \cdot 4 + 4 x}$

Step 3.3.1.2

Rewrite the expression.

$\frac{1 + 4 (\frac{1}{- 4})}{4 \cdot 4 + 4 x}$

Step 3.3.2

Cancel the common factor of $4$ .

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Step 3.3.2.1

Factor $4$ out of $- 4$ .

$\frac{1 + 4 \frac{1}{4 (- 1)}}{4 \cdot 4 + 4 x}$

Step 3.3.2.2

Cancel the common factor.

$\frac{1 + 4 \frac{1}{4 \cdot - 1}}{4 \cdot 4 + 4 x}$

Step 3.3.2.3

Rewrite the expression.

$\frac{1 + \frac{1}{- 1}}{4 \cdot 4 + 4 x}$

Step 3.4

Move the negative one from the denominator of $\frac{1}{- 1}$ .

$\frac{1 - 1 \cdot 1}{4 \cdot 4 + 4 x}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $- 1$ by $1$ .

$\frac{1 - 1}{4 \cdot 4 + 4 x}$

Step 3.5.2

Subtract $1$ from $1$ .

$\frac{0}{4 \cdot 4 + 4 x}$

Step 3.6

Simplify the denominator.

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Step 3.6.1

Multiply $4$ by $4$ .

$\frac{0}{16 + 4 x}$

Step 3.6.2

Factor $4$ out of $16 + 4 x$ .

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Step 3.6.2.1

Factor $4$ out of $16$ .

$\frac{0}{4 \cdot 4 + 4 x}$

Step 3.6.2.2

Factor $4$ out of $4 \cdot 4 + 4 x$ .

$\frac{0}{4 (4 + x)}$

Step 3.7

Cancel the common factor of $0$ and $4$ .

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Step 3.7.1

Factor $4$ out of $0$ .

$\frac{4 (0)}{4 (4 + x)}$

Step 3.7.2

Cancel the common factors.

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Step 3.7.2.1

Cancel the common factor.

$\frac{4 \cdot 0}{4 (4 + x)}$

Step 3.7.2.2

Rewrite the expression.

$\frac{0}{4 + x}$

Step 3.8

Divide $0$ by $4 + x$ .

$0$