Calculus Examples

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

Step 1

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

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

Step 2

Simplify the answer.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Combine.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Simplify by cancelling.

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Step 2.3.1

Cancel the common factor of $4$ .

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Step 2.3.1.1

Factor $4$ out of $4 x$ .

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

Step 2.3.1.2

Cancel the common factor.

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

Step 2.3.1.3

Rewrite the expression.

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

Step 2.3.2

Cancel the common factor of $x$ .

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Step 2.3.2.1

Factor $x$ out of $4 x$ .

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

Step 2.3.2.2

Cancel the common factor.

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

Step 2.3.2.3

Rewrite the expression.

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

Step 2.4

Simplify the denominator.

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Step 2.4.1

Multiply $4$ by $4$ .

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

Step 2.4.2

Multiply $x$ by $x$ by adding the exponents.

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Step 2.4.2.1

Move $x$ .

$\frac{x + 4}{16 x + 4 (x \cdot x)}$

Step 2.4.2.2

Multiply $x$ by $x$ .

$\frac{x + 4}{16 x + 4 x^{2}}$

Step 2.4.3

Factor $4 x$ out of $16 x + 4 x^{2}$ .

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Step 2.4.3.1

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

$\frac{x + 4}{4 x (4) + 4 x^{2}}$

Step 2.4.3.2

Factor $4 x$ out of $4 x^{2}$ .

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

Step 2.4.3.3

Factor $4 x$ out of $4 x (4) + 4 x (x)$ .

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

Step 2.5

Cancel the common factor of $x + 4$ and $4 + x$ .

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Step 2.5.1

Reorder terms.

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

Step 2.5.2

Cancel the common factor.

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

Step 2.5.3

Rewrite the expression.

$\frac{1}{4 x}$