Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Evaluate the limits by plugging in $4$ for all occurrences of $x$ .

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Step 8.1

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

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

Step 8.2

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

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

Step 9

Simplify the answer.

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Step 9.1

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

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Step 9.1.1

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

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

Step 9.1.2

Combine.

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

Step 9.2

Apply the distributive property.

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

Step 9.3

Simplify by cancelling.

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Step 9.3.1

Cancel the common factor of $4$ .

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Step 9.3.1.1

Cancel the common factor.

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

Step 9.3.1.2

Rewrite the expression.

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

Step 9.3.2

Cancel the common factor of $4$ .

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Step 9.3.2.1

Cancel the common factor.

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

Step 9.3.2.2

Rewrite the expression.

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

Step 9.4

Add $1$ and $1$ .

$\frac{2}{4 \cdot 4 + 4 \cdot 4}$

Step 9.5

Simplify the denominator.

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Step 9.5.1

Multiply $4$ by $4$ .

$\frac{2}{16 + 4 \cdot 4}$

Step 9.5.2

Multiply $4$ by $4$ .

$\frac{2}{16 + 16}$

Step 9.5.3

Add $16$ and $16$ .

$\frac{2}{32}$

Step 9.6

Cancel the common factor of $2$ and $32$ .

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Step 9.6.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{32}$

Step 9.6.2

Cancel the common factors.

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Step 9.6.2.1

Factor $2$ out of $32$ .

$\frac{2 \cdot 1}{2 \cdot 16}$

Step 9.6.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 16}$

Step 9.6.2.3

Rewrite the expression.

$\frac{1}{16}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{16}$

Decimal Form:

$0.0625$