Calculus Examples

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

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $5$ .

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

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $5$ .

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

Step 3

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

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

Step 4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $5$ .

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

Step 5

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

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

Step 6

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

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

Step 7

Evaluate the limits by plugging in $5$ for all occurrences of $x$ .

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Step 7.1

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

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

Step 7.2

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

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

Step 8

Simplify the answer.

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Step 8.1

Cancel the common factor of $5 - 1 \cdot 5$ and $5^{2} - 1 \cdot 5$ .

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Step 8.1.1

Rewrite $5$ as $- 1 (- 5)$ .

$\frac{- 1 (- 5) - 1 \cdot 5}{5^{2} - 1 \cdot 5}$

Step 8.1.2

Factor $- 1$ out of $- 1 \cdot 5$ .

$\frac{- 1 (- 5) - 1 (5)}{5^{2} - 1 \cdot 5}$

Step 8.1.3

Factor $- 1$ out of $- 1 (- 5) - 1 (5)$ .

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

Step 8.1.4

Factor $5$ out of $- 1 (- 5 + 5)$ .

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

Step 8.1.5

Cancel the common factors.

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Step 8.1.5.1

Factor $5$ out of $5^{2}$ .

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

Step 8.1.5.2

Factor $5$ out of $- 1 \cdot 5$ .

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

Step 8.1.5.3

Factor $5$ out of $5 \cdot 5 + 5 \cdot - 1$ .

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

Step 8.1.5.4

Cancel the common factor.

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

Step 8.1.5.5

Rewrite the expression.

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

Step 8.2

Add $- 1$ and $1$ .

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

Step 8.3

Subtract $1$ from $5$ .

$\frac{- 1 \cdot 0}{4}$

Step 8.4

Multiply $- 1$ by $0$ .

$\frac{0}{4}$

Step 8.5

Divide $0$ by $4$ .

$0$