Calculus Examples

$lim_{x \to - 8} \frac{\sqrt{x^{2} - 9}}{2 x - 6}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- 8$ .

$\frac{lim_{x \to - 8} \sqrt{x^{2} - 9}}{lim_{x \to - 8} 2 x - 6}$

Step 2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to - 8} x^{2} - 9}}{lim_{x \to - 8} 2 x - 6}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 8$ .

$\frac{\sqrt{lim_{x \to - 8} x^{2} - lim_{x \to - 8} 9}}{lim_{x \to - 8} 2 x - 6}$

Step 4

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

$\frac{\sqrt{{(lim_{x \to - 8} x)}^{2} - lim_{x \to - 8} 9}}{lim_{x \to - 8} 2 x - 6}$

Step 5

Evaluate the limit of $9$ which is constant as $x$ approaches $- 8$ .

$\frac{\sqrt{{(lim_{x \to - 8} x)}^{2} - 1 \cdot 9}}{lim_{x \to - 8} 2 x - 6}$

Step 6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 8$ .

$\frac{\sqrt{{(lim_{x \to - 8} x)}^{2} - 1 \cdot 9}}{lim_{x \to - 8} 2 x - lim_{x \to - 8} 6}$

Step 7

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sqrt{{(lim_{x \to - 8} x)}^{2} - 1 \cdot 9}}{2 lim_{x \to - 8} x - lim_{x \to - 8} 6}$

Step 8

Evaluate the limit of $6$ which is constant as $x$ approaches $- 8$ .

$\frac{\sqrt{{(lim_{x \to - 8} x)}^{2} - 1 \cdot 9}}{2 lim_{x \to - 8} x - 1 \cdot 6}$

Step 9

Evaluate the limits by plugging in $- 8$ for all occurrences of $x$ .

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Step 9.1

Evaluate the limit of $x$ by plugging in $- 8$ for $x$ .

$\frac{\sqrt{{(- 8)}^{2} - 1 \cdot 9}}{2 lim_{x \to - 8} x - 1 \cdot 6}$

Step 9.2

Evaluate the limit of $x$ by plugging in $- 8$ for $x$ .

$\frac{\sqrt{{(- 8)}^{2} - 1 \cdot 9}}{2 \cdot - 8 - 1 \cdot 6}$

Step 10

Simplify the answer.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Raise $- 8$ to the power of $2$ .

$\frac{\sqrt{64 - 1 \cdot 9}}{2 \cdot - 8 - 1 \cdot 6}$

Step 10.1.2

Multiply $- 1$ by $9$ .

$\frac{\sqrt{64 - 9}}{2 \cdot - 8 - 1 \cdot 6}$

Step 10.1.3

Subtract $9$ from $64$ .

$\frac{\sqrt{55}}{2 \cdot - 8 - 1 \cdot 6}$

Step 10.2

Simplify the denominator.

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Step 10.2.1

Multiply $2$ by $- 8$ .

$\frac{\sqrt{55}}{- 16 - 1 \cdot 6}$

Step 10.2.2

Multiply $- 1$ by $6$ .

$\frac{\sqrt{55}}{- 16 - 6}$

Step 10.2.3

Subtract $6$ from $- 16$ .

$\frac{\sqrt{55}}{- 22}$

Step 10.3

Move the negative in front of the fraction.

$- \frac{\sqrt{55}}{22}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$- \frac{\sqrt{55}}{22}$

Decimal Form:

$- 0.33709993 \dots$