Calculus Examples

$\frac{\frac{lim_{x \to 4} x}{x + 1} - \frac{4}{5}}{x - 4}$

Step 1

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

$\frac{\frac{4}{x + 1} - \frac{4}{5}}{x - 4}$

Step 2

Simplify the answer.

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Step 2.1

Multiply the numerator and denominator of the fraction by $(x + 1) \cdot 5$ .

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Step 2.1.1

Multiply $\frac{\frac{4}{x + 1} - \frac{4}{5}}{x - 4}$ by $\frac{(x + 1) \cdot 5}{(x + 1) \cdot 5}$ .

$\frac{(x + 1) \cdot 5}{(x + 1) \cdot 5} \cdot \frac{\frac{4}{x + 1} - \frac{4}{5}}{x - 4}$

Step 2.1.2

Combine.

$\frac{(x + 1) \cdot 5 (\frac{4}{x + 1} - \frac{4}{5})}{(x + 1) \cdot 5 (x - 4)}$

Step 2.2

Apply the distributive property.

$\frac{(x + 1) \cdot 5 \frac{4}{x + 1} + (x + 1) \cdot 5 (- \frac{4}{5})}{(x + 1) \cdot 5 x + (x + 1) \cdot 5 \cdot - 4}$

Step 2.3

Simplify by cancelling.

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Step 2.3.1

Cancel the common factor of $x + 1$ .

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Step 2.3.1.1

Factor $x + 1$ out of $(x + 1) \cdot 5$ .

$\frac{(x + 1) (5) \frac{4}{x + 1} + (x + 1) \cdot 5 (- \frac{4}{5})}{(x + 1) \cdot 5 x + (x + 1) \cdot 5 \cdot - 4}$

Step 2.3.1.2

Cancel the common factor.

$\frac{(x + 1) \cdot 5 \frac{4}{x + 1} + (x + 1) \cdot 5 (- \frac{4}{5})}{(x + 1) \cdot 5 x + (x + 1) \cdot 5 \cdot - 4}$

Step 2.3.1.3

Rewrite the expression.

$\frac{5 \cdot 4 + (x + 1) \cdot 5 (- \frac{4}{5})}{(x + 1) \cdot 5 x + (x + 1) \cdot 5 \cdot - 4}$

Step 2.3.2

Multiply $5$ by $4$ .

$\frac{20 + (x + 1) \cdot 5 (- \frac{4}{5})}{(x + 1) \cdot 5 x + (x + 1) \cdot 5 \cdot - 4}$

Step 2.3.3

Cancel the common factor of $5$ .

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Step 2.3.3.1

Move the leading negative in $- \frac{4}{5}$ into the numerator.

$\frac{20 + (x + 1) \cdot 5 \frac{- 4}{5}}{(x + 1) \cdot 5 x + (x + 1) \cdot 5 \cdot - 4}$

Step 2.3.3.2

Factor $5$ out of $(x + 1) \cdot 5$ .

$\frac{20 + 5 \cdot (x + 1) \frac{- 4}{5}}{(x + 1) \cdot 5 x + (x + 1) \cdot 5 \cdot - 4}$

Step 2.3.3.3

Cancel the common factor.

$\frac{20 + 5 \cdot (x + 1) \frac{- 4}{5}}{(x + 1) \cdot 5 x + (x + 1) \cdot 5 \cdot - 4}$

Step 2.3.3.4

Rewrite the expression.

$\frac{20 + (x + 1) \cdot - 4}{(x + 1) \cdot 5 x + (x + 1) \cdot 5 \cdot - 4}$

Step 2.4

Simplify the numerator.

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Step 2.4.1

Factor $4$ out of $20 + (x + 1) \cdot - 4$ .

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Step 2.4.1.1

Factor $4$ out of $20$ .

$\frac{4 (5) + (x + 1) \cdot - 4}{(x + 1) \cdot 5 x + (x + 1) \cdot 5 \cdot - 4}$

Step 2.4.1.2

Factor $4$ out of $(x + 1) \cdot - 4$ .

$\frac{4 (5) + 4 ((x + 1) \cdot - 1)}{(x + 1) \cdot 5 x + (x + 1) \cdot 5 \cdot - 4}$

Step 2.4.1.3

Factor $4$ out of $4 (5) + 4 ((x + 1) \cdot - 1)$ .

$\frac{4 (5 + (x + 1) \cdot - 1)}{(x + 1) \cdot 5 x + (x + 1) \cdot 5 \cdot - 4}$

Step 2.4.2

Apply the distributive property.

$\frac{4 (5 + x \cdot - 1 + 1 \cdot - 1)}{(x + 1) \cdot 5 x + (x + 1) \cdot 5 \cdot - 4}$

Step 2.4.3

Move $- 1$ to the left of $x$ .

$\frac{4 (5 - 1 \cdot x + 1 \cdot - 1)}{(x + 1) \cdot 5 x + (x + 1) \cdot 5 \cdot - 4}$

Step 2.4.4

Multiply $- 1$ by $1$ .

$\frac{4 (5 - 1 \cdot x - 1)}{(x + 1) \cdot 5 x + (x + 1) \cdot 5 \cdot - 4}$

Step 2.4.5

Rewrite $- 1 x$ as $- x$ .

$\frac{4 (5 - x - 1)}{(x + 1) \cdot 5 x + (x + 1) \cdot 5 \cdot - 4}$

Step 2.4.6

Subtract $1$ from $5$ .

$\frac{4 (- x + 4)}{(x + 1) \cdot 5 x + (x + 1) \cdot 5 \cdot - 4}$

Step 2.5

Factor $(x + 1) \cdot 5$ out of $(x + 1) \cdot 5 x + (x + 1) \cdot 5 \cdot - 4$ .

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Step 2.5.1

Factor $(x + 1) \cdot 5$ out of $(x + 1) \cdot 5 x$ .

$\frac{4 (- x + 4)}{(x + 1) \cdot 5 (x) + (x + 1) \cdot 5 \cdot - 4}$

Step 2.5.2

Factor $(x + 1) \cdot 5$ out of $(x + 1) \cdot 5 \cdot - 4$ .

$\frac{4 (- x + 4)}{(x + 1) \cdot 5 (x) + (x + 1) \cdot 5 (- 4)}$

Step 2.5.3

Factor $(x + 1) \cdot 5$ out of $(x + 1) \cdot 5 (x) + (x + 1) \cdot 5 (- 4)$ .

$\frac{4 (- x + 4)}{(x + 1) \cdot 5 (x - 4)}$

Step 2.6

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

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Step 2.6.1

Factor $- 1$ out of $- x$ .

$\frac{4 (- (x) + 4)}{(x + 1) \cdot 5 (x - 4)}$

Step 2.6.2

Rewrite $4$ as $- 1 (- 4)$ .

$\frac{4 (- (x) - 1 (- 4))}{(x + 1) \cdot 5 (x - 4)}$

Step 2.6.3

Factor $- 1$ out of $- (x) - 1 (- 4)$ .

$\frac{4 (- (x - 4))}{(x + 1) \cdot 5 (x - 4)}$

Step 2.6.4

Rewrite $- (x - 4)$ as $- 1 (x - 4)$ .

$\frac{4 (- 1 (x - 4))}{(x + 1) \cdot 5 (x - 4)}$

Step 2.6.5

Cancel the common factor.

$\frac{4 (- 1 (x - 4))}{(x + 1) \cdot 5 (x - 4)}$

Step 2.6.6

Rewrite the expression.

$\frac{4 \cdot (- 1)}{(x + 1) \cdot 5}$

Step 2.7

Multiply $4$ by $- 1$ .

$\frac{- 4}{(x + 1) \cdot 5}$

Step 2.8

Move $5$ to the left of $x + 1$ .

$\frac{- 4}{5 \cdot (x + 1)}$

Step 2.9

Move the negative in front of the fraction.

$- \frac{4}{5 (x + 1)}$