Calculus Examples

$lim_{x \to - 5} \frac{x^{2} - 9}{5 - x}$

Step 1

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

$\frac{lim_{x \to - 5} x^{2} - 9}{lim_{x \to - 5} 5 - x}$

Step 2

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

$\frac{lim_{x \to - 5} x^{2} - lim_{x \to - 5} 9}{lim_{x \to - 5} 5 - x}$

Step 3

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

$\frac{{(lim_{x \to - 5} x)}^{2} - lim_{x \to - 5} 9}{lim_{x \to - 5} 5 - x}$

Step 4

Evaluate the limit of $9$ which is constant as $x$ approaches $- 5$ .

$\frac{{(lim_{x \to - 5} x)}^{2} - 1 \cdot 9}{lim_{x \to - 5} 5 - x}$

Step 5

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

$\frac{{(lim_{x \to - 5} x)}^{2} - 1 \cdot 9}{lim_{x \to - 5} 5 - lim_{x \to - 5} x}$

Step 6

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

$\frac{{(lim_{x \to - 5} x)}^{2} - 1 \cdot 9}{5 - lim_{x \to - 5} x}$

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 $x$ .

$\frac{{(- 5)}^{2} - 1 \cdot 9}{5 - lim_{x \to - 5} x}$

Step 7.2

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

$\frac{{(- 5)}^{2} - 1 \cdot 9}{5 + 5}$

Step 8

Simplify the answer.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $- 5$ to the power of $2$ .

$\frac{25 - 1 \cdot 9}{5 + 5}$

Step 8.1.2

Multiply $- 1$ by $9$ .

$\frac{25 - 9}{5 + 5}$

Step 8.1.3

Subtract $9$ from $25$ .

$\frac{16}{5 + 5}$

Step 8.2

Add $5$ and $5$ .

$\frac{16}{10}$

Step 8.3

Cancel the common factor of $16$ and $10$ .

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Step 8.3.1

Factor $2$ out of $16$ .

$\frac{2 (8)}{10}$

Step 8.3.2

Cancel the common factors.

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Step 8.3.2.1

Factor $2$ out of $10$ .

$\frac{2 \cdot 8}{2 \cdot 5}$

Step 8.3.2.2

Cancel the common factor.

$\frac{2 \cdot 8}{2 \cdot 5}$

Step 8.3.2.3

Rewrite the expression.

$\frac{8}{5}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{8}{5}$

Decimal Form:

$1.6$