Calculus Examples

$\frac{lim_{x \to 5} x - 5}{x^{2} - 25}$

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 $5$ .

$\frac{lim_{x \to 5} x - lim_{x \to 5} 5}{x^{2} - 25}$

Step 1.2

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

$\frac{lim_{x \to 5} x - 1 \cdot 5}{x^{2} - 25}$

Step 2

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

$\frac{5 - 1 \cdot 5}{x^{2} - 25}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Multiply $- 1$ by $5$ .

$\frac{5 - 5}{x^{2} - 25}$

Step 3.1.2

Subtract $5$ from $5$ .

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

Step 3.2

Simplify the denominator.

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Step 3.2.1

Rewrite $25$ as $5^{2}$ .

$\frac{0}{x^{2} - 5^{2}}$

Step 3.2.2

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 $b = 5$ .

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

Step 3.3

Divide $0$ by $(x + 5) (x - 5)$ .

$0$