Calculus Examples

$lim_{x \to 0} (\frac{\frac{1}{x + 5}}{x}) - (\frac{\frac{1}{5}}{x})$

Step 1

Combine fractions using a common denominator.

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Step 1.1

Combine the numerators over the common denominator.

$lim_{x \to 0} \frac{\frac{1}{x + 5} - \frac{1}{5}}{x}$

Step 1.2

Simplify the numerator.

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Step 1.2.1

To write $\frac{1}{x + 5}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$lim_{x \to 0} \frac{\frac{1}{x + 5} \cdot \frac{5}{5} - \frac{1}{5}}{x}$

Step 1.2.2

To write $- \frac{1}{5}$ as a fraction with a common denominator, multiply by $\frac{x + 5}{x + 5}$ .

$lim_{x \to 0} \frac{\frac{1}{x + 5} \cdot \frac{5}{5} - \frac{1}{5} \cdot \frac{x + 5}{x + 5}}{x}$

Step 1.2.3

Write each expression with a common denominator of $(x + 5) \cdot 5$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.2.3.1

Multiply $\frac{1}{x + 5}$ by $\frac{5}{5}$ .

$lim_{x \to 0} \frac{\frac{5}{(x + 5) \cdot 5} - \frac{1}{5} \cdot \frac{x + 5}{x + 5}}{x}$

Step 1.2.3.2

Multiply $\frac{1}{5}$ by $\frac{x + 5}{x + 5}$ .

$lim_{x \to 0} \frac{\frac{5}{(x + 5) \cdot 5} - \frac{x + 5}{5 (x + 5)}}{x}$

Step 1.2.3.3

Reorder the factors of $(x + 5) \cdot 5$ .

$lim_{x \to 0} \frac{\frac{5}{5 (x + 5)} - \frac{x + 5}{5 (x + 5)}}{x}$

Step 1.2.4

Combine the numerators over the common denominator.

$lim_{x \to 0} \frac{\frac{5 - (x + 5)}{5 (x + 5)}}{x}$

Step 1.2.5

Rewrite $\frac{5 - (x + 5)}{5 (x + 5)}$ in a factored form.

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Step 1.2.5.1

Apply the distributive property.

$lim_{x \to 0} \frac{\frac{5 - x - 1 \cdot 5}{5 (x + 5)}}{x}$

Step 1.2.5.2

Multiply $- 1$ by $5$ .

$lim_{x \to 0} \frac{\frac{5 - x - 5}{5 (x + 5)}}{x}$

Step 1.2.5.3

Subtract $5$ from $5$ .

$lim_{x \to 0} \frac{\frac{- x + 0}{5 (x + 5)}}{x}$

Step 1.2.5.4

Add $- x$ and $0$ .

$lim_{x \to 0} \frac{\frac{- x}{5 (x + 5)}}{x}$

Step 1.2.6

Move the negative in front of the fraction.

$lim_{x \to 0} \frac{- \frac{x}{5 (x + 5)}}{x}$

Step 1.3

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 0} - \frac{x}{5 (x + 5)} \cdot \frac{1}{x}$

Step 1.4

Cancel the common factor of $x$ .

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Step 1.4.1

Move the leading negative in $- \frac{x}{5 (x + 5)}$ into the numerator.

$lim_{x \to 0} \frac{- x}{5 (x + 5)} \cdot \frac{1}{x}$

Step 1.4.2

Factor $x$ out of $- x$ .

$lim_{x \to 0} \frac{x \cdot - 1}{5 (x + 5)} \cdot \frac{1}{x}$

Step 1.4.3

Cancel the common factor.

$lim_{x \to 0} \frac{x \cdot - 1}{5 (x + 5)} \cdot \frac{1}{x}$

Step 1.4.4

Rewrite the expression.

$lim_{x \to 0} \frac{- 1}{5 (x + 5)}$

Step 1.5

Move the negative in front of the fraction.

$lim_{x \to 0} - \frac{1}{5 (x + 5)}$

Step 2

Evaluate the limit.

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Step 2.1

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

$- lim_{x \to 0} \frac{1}{5 (x + 5)}$

Step 2.2

Move the term $\frac{1}{5}$ outside of the limit because it is constant with respect to $x$ .

$- \frac{1}{5} lim_{x \to 0} \frac{1}{x + 5}$

Step 2.3

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $0$ .

$- \frac{1}{5} \cdot \frac{lim_{x \to 0} 1}{lim_{x \to 0} x + 5}$

Step 2.4

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$- \frac{1}{5} \cdot \frac{1}{lim_{x \to 0} x + 5}$

Step 2.5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$- \frac{1}{5} \cdot \frac{1}{lim_{x \to 0} x + lim_{x \to 0} 5}$

Step 2.6

Evaluate the limit of $5$ which is constant as $x$ approaches $0$ .

$- \frac{1}{5} \cdot \frac{1}{lim_{x \to 0} x + 5}$

Step 3

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$- \frac{1}{5} \cdot \frac{1}{0 + 5}$

Step 4

Simplify the answer.

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Step 4.1

Add $0$ and $5$ .

$- \frac{1}{5} \cdot \frac{1}{5}$

Step 4.2

Multiply $- \frac{1}{5} \cdot \frac{1}{5}$ .

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Step 4.2.1

Multiply $\frac{1}{5}$ by $\frac{1}{5}$ .

$- \frac{1}{5 \cdot 5}$

Step 4.2.2

Multiply $5$ by $5$ .

$- \frac{1}{25}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{25}$

Decimal Form:

$- 0.04$