Calculus Examples

$lim_{x \to 4.1} \frac{(\frac{x}{x + 1}) - (\frac{4}{5})}{x - 4}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $4.1$ .

$\frac{lim_{x \to 4.1} (\frac{x}{x + 1}) - (\frac{4}{5})}{lim_{x \to 4.1} x - 4}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4.1$ .

$\frac{lim_{x \to 4.1} \frac{x}{x + 1} - lim_{x \to 4.1} \frac{4}{5}}{lim_{x \to 4.1} x - 4}$

Step 3

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $4.1$ .

$\frac{\frac{lim_{x \to 4.1} x}{lim_{x \to 4.1} x + 1} - lim_{x \to 4.1} \frac{4}{5}}{lim_{x \to 4.1} x - 4}$

Step 4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4.1$ .

$\frac{\frac{lim_{x \to 4.1} x}{lim_{x \to 4.1} x + lim_{x \to 4.1} 1} - lim_{x \to 4.1} \frac{4}{5}}{lim_{x \to 4.1} x - 4}$

Step 5

Evaluate the limit of $1$ which is constant as $x$ approaches $4.1$ .

$\frac{\frac{lim_{x \to 4.1} x}{lim_{x \to 4.1} x + 1} - lim_{x \to 4.1} \frac{4}{5}}{lim_{x \to 4.1} x - 4}$

Step 6

Evaluate the limit of $\frac{4}{5}$ which is constant as $x$ approaches $4.1$ .

$\frac{\frac{lim_{x \to 4.1} x}{lim_{x \to 4.1} x + 1} - \frac{4}{5}}{lim_{x \to 4.1} x - 4}$

Step 7

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4.1$ .

$\frac{\frac{lim_{x \to 4.1} x}{lim_{x \to 4.1} x + 1} - \frac{4}{5}}{lim_{x \to 4.1} x - lim_{x \to 4.1} 4}$

Step 8

Evaluate the limit of $4$ which is constant as $x$ approaches $4.1$ .

$\frac{\frac{lim_{x \to 4.1} x}{lim_{x \to 4.1} x + 1} - \frac{4}{5}}{lim_{x \to 4.1} x - 1 \cdot 4}$

Step 9

Evaluate the limits by plugging in $4.1$ for all occurrences of $x$ .

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Step 9.1

Evaluate the limit of $x$ by plugging in $4.1$ for $x$ .

$\frac{\frac{4.1}{lim_{x \to 4.1} x + 1} - \frac{4}{5}}{lim_{x \to 4.1} x - 1 \cdot 4}$

Step 9.2

Evaluate the limit of $x$ by plugging in $4.1$ for $x$ .

$\frac{\frac{4.1}{4.1 + 1} - \frac{4}{5}}{lim_{x \to 4.1} x - 1 \cdot 4}$

Step 9.3

Evaluate the limit of $x$ by plugging in $4.1$ for $x$ .

$\frac{\frac{4.1}{4.1 + 1} - \frac{4}{5}}{4.1 - 1 \cdot 4}$

Step 10

Simplify the answer.

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Step 10.1

Multiply the numerator and denominator of the fraction by $(4.1 + 1) \cdot 5$ .

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Step 10.1.1

Multiply $\frac{\frac{4.1}{4.1 + 1} - \frac{4}{5}}{4.1 - 1 \cdot 4}$ by $\frac{(4.1 + 1) \cdot 5}{(4.1 + 1) \cdot 5}$ .

$\frac{(4.1 + 1) \cdot 5}{(4.1 + 1) \cdot 5} \cdot \frac{\frac{4.1}{4.1 + 1} - \frac{4}{5}}{4.1 - 1 \cdot 4}$

Step 10.1.2

Combine.

$\frac{(4.1 + 1) \cdot 5 (\frac{4.1}{4.1 + 1} - \frac{4}{5})}{(4.1 + 1) \cdot 5 (4.1 - 1 \cdot 4)}$

Step 10.2

Apply the distributive property.

$\frac{(4.1 + 1) \cdot 5 \frac{4.1}{4.1 + 1} + (4.1 + 1) \cdot 5 (- \frac{4}{5})}{(4.1 + 1) \cdot 5 \cdot 4.1 + (4.1 + 1) \cdot 5 (- 1 \cdot 4)}$

Step 10.3

Simplify by cancelling.

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Step 10.3.1

Cancel the common factor of $4.1 + 1$ .

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Step 10.3.1.1

Factor $4.1 + 1$ out of $(4.1 + 1) \cdot 5$ .

$\frac{(4.1 + 1) (5) \frac{4.1}{4.1 + 1} + (4.1 + 1) \cdot 5 (- \frac{4}{5})}{(4.1 + 1) \cdot 5 \cdot 4.1 + (4.1 + 1) \cdot 5 (- 1 \cdot 4)}$

Step 10.3.1.2

Cancel the common factor.

$\frac{(4.1 + 1) \cdot 5 \frac{4.1}{4.1 + 1} + (4.1 + 1) \cdot 5 (- \frac{4}{5})}{(4.1 + 1) \cdot 5 \cdot 4.1 + (4.1 + 1) \cdot 5 (- 1 \cdot 4)}$

Step 10.3.1.3

Rewrite the expression.

$\frac{5 \cdot 4.1 + (4.1 + 1) \cdot 5 (- \frac{4}{5})}{(4.1 + 1) \cdot 5 \cdot 4.1 + (4.1 + 1) \cdot 5 (- 1 \cdot 4)}$

Step 10.3.2

Multiply $5$ by $4.1$ .

$\frac{20.5 + (4.1 + 1) \cdot 5 (- \frac{4}{5})}{(4.1 + 1) \cdot 5 \cdot 4.1 + (4.1 + 1) \cdot 5 (- 1 \cdot 4)}$

Step 10.3.3

Cancel the common factor of $5$ .

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Step 10.3.3.1

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

$\frac{20.5 + (4.1 + 1) \cdot 5 \frac{- 4}{5}}{(4.1 + 1) \cdot 5 \cdot 4.1 + (4.1 + 1) \cdot 5 (- 1 \cdot 4)}$

Step 10.3.3.2

Factor $5$ out of $(4.1 + 1) \cdot 5$ .

$\frac{20.5 + 5 \cdot (4.1 + 1) \frac{- 4}{5}}{(4.1 + 1) \cdot 5 \cdot 4.1 + (4.1 + 1) \cdot 5 (- 1 \cdot 4)}$

Step 10.3.3.3

Cancel the common factor.

$\frac{20.5 + 5 \cdot (4.1 + 1) \frac{- 4}{5}}{(4.1 + 1) \cdot 5 \cdot 4.1 + (4.1 + 1) \cdot 5 (- 1 \cdot 4)}$

Step 10.3.3.4

Rewrite the expression.

$\frac{20.5 + (4.1 + 1) \cdot - 4}{(4.1 + 1) \cdot 5 \cdot 4.1 + (4.1 + 1) \cdot 5 (- 1 \cdot 4)}$

Step 10.4

Simplify the numerator.

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Step 10.4.1

Add $4.1$ and $1$ .

$\frac{20.5 + 5.1 \cdot - 4}{(4.1 + 1) \cdot 5 \cdot 4.1 + (4.1 + 1) \cdot 5 (- 1 \cdot 4)}$

Step 10.4.2

Multiply $5.1$ by $- 4$ .

$\frac{20.5 - 20.4}{(4.1 + 1) \cdot 5 \cdot 4.1 + (4.1 + 1) \cdot 5 (- 1 \cdot 4)}$

Step 10.4.3

Subtract $20.4$ from $20.5$ .

$\frac{0.1}{(4.1 + 1) \cdot 5 \cdot 4.1 + (4.1 + 1) \cdot 5 (- 1 \cdot 4)}$

Step 10.5

Simplify the denominator.

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Step 10.5.1

Add $4.1$ and $1$ .

$\frac{0.1}{5.1 \cdot 5 \cdot 4.1 + (4.1 + 1) \cdot 5 (- 1 \cdot 4)}$

Step 10.5.2

Multiply $5.1 \cdot 5 \cdot 4.1$ .

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Step 10.5.2.1

Multiply $5.1$ by $5$ .

$\frac{0.1}{25.5 \cdot 4.1 + (4.1 + 1) \cdot 5 (- 1 \cdot 4)}$

Step 10.5.2.2

Multiply $25.5$ by $4.1$ .

$\frac{0.1}{104.55 + (4.1 + 1) \cdot 5 (- 1 \cdot 4)}$

Step 10.5.3

Add $4.1$ and $1$ .

$\frac{0.1}{104.55 + 5.1 \cdot 5 (- 1 \cdot 4)}$

Step 10.5.4

Multiply $5.1 \cdot 5 (- 1 \cdot 4)$ .

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Step 10.5.4.1

Multiply $5.1$ by $5$ .

$\frac{0.1}{104.55 + 25.5 (- 1 \cdot 4)}$

Step 10.5.4.2

Multiply $- 1$ by $4$ .

$\frac{0.1}{104.55 + 25.5 \cdot - 4}$

Step 10.5.4.3

Multiply $25.5$ by $- 4$ .

$\frac{0.1}{104.55 - 102}$

Step 10.5.5

Subtract $102$ from $104.55$ .

$\frac{0.1}{2.55}$

Step 10.6

Divide $0.1$ by $2.55$ .

$0.03921568$