Calculus Examples

$\frac{lim_{x \to 2} \frac{1}{x + 3} - \frac{1}{5}}{x^{2} - 4}$

Step 1

Evaluate the limit.

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Step 1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{lim_{x \to 2} \frac{1}{x + 3} - lim_{x \to 2} \frac{1}{5}}{x^{2} - 4}$

Step 1.2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $2$ .

$\frac{\frac{lim_{x \to 2} 1}{lim_{x \to 2} x + 3} - lim_{x \to 2} \frac{1}{5}}{x^{2} - 4}$

Step 1.3

Evaluate the limit of $1$ which is constant as $x$ approaches $2$ .

$\frac{\frac{1}{lim_{x \to 2} x + 3} - lim_{x \to 2} \frac{1}{5}}{x^{2} - 4}$

Step 1.4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{\frac{1}{lim_{x \to 2} x + lim_{x \to 2} 3} - lim_{x \to 2} \frac{1}{5}}{x^{2} - 4}$

Step 1.5

Evaluate the limit of $3$ which is constant as $x$ approaches $2$ .

$\frac{\frac{1}{lim_{x \to 2} x + 3} - lim_{x \to 2} \frac{1}{5}}{x^{2} - 4}$

Step 1.6

Evaluate the limit of $\frac{1}{5}$ which is constant as $x$ approaches $2$ .

$\frac{\frac{1}{lim_{x \to 2} x + 3} - \frac{1}{5}}{x^{2} - 4}$

Step 2

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{\frac{1}{2 + 3} - \frac{1}{5}}{x^{2} - 4}$

Step 3

Simplify the answer.

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Step 3.1

Multiply the numerator and denominator of the fraction by $(2 + 3) \cdot 5$ .

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Step 3.1.1

Multiply $\frac{\frac{1}{2 + 3} - \frac{1}{5}}{x^{2} - 4}$ by $\frac{(2 + 3) \cdot 5}{(2 + 3) \cdot 5}$ .

$\frac{(2 + 3) \cdot 5}{(2 + 3) \cdot 5} \cdot \frac{\frac{1}{2 + 3} - \frac{1}{5}}{x^{2} - 4}$

Step 3.1.2

Combine.

$\frac{(2 + 3) \cdot 5 (\frac{1}{2 + 3} - \frac{1}{5})}{(2 + 3) \cdot 5 (x^{2} - 4)}$

Step 3.2

Apply the distributive property.

$\frac{(2 + 3) \cdot 5 \frac{1}{2 + 3} + (2 + 3) \cdot 5 (- \frac{1}{5})}{(2 + 3) \cdot 5 x^{2} + (2 + 3) \cdot 5 \cdot - 4}$

Step 3.3

Simplify by cancelling.

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Step 3.3.1

Cancel the common factor of $2 + 3$ .

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Step 3.3.1.1

Factor $2 + 3$ out of $(2 + 3) \cdot 5$ .

$\frac{(2 + 3) (5) \frac{1}{2 + 3} + (2 + 3) \cdot 5 (- \frac{1}{5})}{(2 + 3) \cdot 5 x^{2} + (2 + 3) \cdot 5 \cdot - 4}$

Step 3.3.1.2

Cancel the common factor.

$\frac{(2 + 3) \cdot 5 \frac{1}{2 + 3} + (2 + 3) \cdot 5 (- \frac{1}{5})}{(2 + 3) \cdot 5 x^{2} + (2 + 3) \cdot 5 \cdot - 4}$

Step 3.3.1.3

Rewrite the expression.

$\frac{5 + (2 + 3) \cdot 5 (- \frac{1}{5})}{(2 + 3) \cdot 5 x^{2} + (2 + 3) \cdot 5 \cdot - 4}$

Step 3.3.2

Cancel the common factor of $5$ .

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Step 3.3.2.1

Move the leading negative in $- \frac{1}{5}$ into the numerator.

$\frac{5 + (2 + 3) \cdot 5 \frac{- 1}{5}}{(2 + 3) \cdot 5 x^{2} + (2 + 3) \cdot 5 \cdot - 4}$

Step 3.3.2.2

Factor $5$ out of $(2 + 3) \cdot 5$ .

$\frac{5 + 5 \cdot (2 + 3) \frac{- 1}{5}}{(2 + 3) \cdot 5 x^{2} + (2 + 3) \cdot 5 \cdot - 4}$

Step 3.3.2.3

Cancel the common factor.

$\frac{5 + 5 \cdot (2 + 3) \frac{- 1}{5}}{(2 + 3) \cdot 5 x^{2} + (2 + 3) \cdot 5 \cdot - 4}$

Step 3.3.2.4

Rewrite the expression.

$\frac{5 + (2 + 3) \cdot - 1}{(2 + 3) \cdot 5 x^{2} + (2 + 3) \cdot 5 \cdot - 4}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Add $2$ and $3$ .

$\frac{5 + 5 \cdot - 1}{(2 + 3) \cdot 5 x^{2} + (2 + 3) \cdot 5 \cdot - 4}$

Step 3.4.2

Multiply $5$ by $- 1$ .

$\frac{5 - 5}{(2 + 3) \cdot 5 x^{2} + (2 + 3) \cdot 5 \cdot - 4}$

Step 3.4.3

Subtract $5$ from $5$ .

$\frac{0}{(2 + 3) \cdot 5 x^{2} + (2 + 3) \cdot 5 \cdot - 4}$

Step 3.5

Simplify the denominator.

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Step 3.5.1

Factor $(2 + 3) \cdot 5$ out of $(2 + 3) \cdot 5 x^{2} + (2 + 3) \cdot 5 \cdot - 4$ .

$\frac{0}{(2 + 3) \cdot 5 (x^{2} - 4)}$

Step 3.5.2

Add $2$ and $3$ .

$\frac{0}{5 \cdot 5 (x^{2} - 4)}$

Step 3.5.3

Multiply $5$ by $5$ .

$\frac{0}{25 (x^{2} - 4)}$

Step 3.5.4

Rewrite $4$ as $2^{2}$ .

$\frac{0}{25 (x^{2} - 2^{2})}$

Step 3.5.5

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

$\frac{0}{25 (x + 2) (x - 2)}$

Step 3.6

Cancel the common factor of $0$ and $25$ .

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Step 3.6.1

Factor $25$ out of $0$ .

$\frac{25 (0)}{25 (x + 2) (x - 2)}$

Step 3.6.2

Cancel the common factors.

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Step 3.6.2.1

Factor $25$ out of $25 (x + 2) (x - 2)$ .

$\frac{25 (0)}{25 ((x + 2) (x - 2))}$

Step 3.6.2.2

Cancel the common factor.

$\frac{25 \cdot 0}{25 ((x + 2) (x - 2))}$

Step 3.6.2.3

Rewrite the expression.

$\frac{0}{(x + 2) (x - 2)}$

Step 3.7

Divide $0$ by $(x + 2) (x - 2)$ .

$0$