Calculus Examples

$lim_{x \to 6} \sqrt{\frac{x^{2} - 9}{x - 3}}$

Step 1

Move the limit under the radical sign.

$\sqrt{lim_{x \to 6} \frac{x^{2} - 9}{x - 3}}$

Step 2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $6$ .

$\sqrt{\frac{lim_{x \to 6} x^{2} - 9}{lim_{x \to 6} x - 3}}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $6$ .

$\sqrt{\frac{lim_{x \to 6} x^{2} - lim_{x \to 6} 9}{lim_{x \to 6} x - 3}}$

Step 4

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

$\sqrt{\frac{{(lim_{x \to 6} x)}^{2} - lim_{x \to 6} 9}{lim_{x \to 6} x - 3}}$

Step 5

Evaluate the limit of $9$ which is constant as $x$ approaches $6$ .

$\sqrt{\frac{{(lim_{x \to 6} x)}^{2} - 1 \cdot 9}{lim_{x \to 6} x - 3}}$

Step 6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $6$ .

$\sqrt{\frac{{(lim_{x \to 6} x)}^{2} - 1 \cdot 9}{lim_{x \to 6} x - lim_{x \to 6} 3}}$

Step 7

Evaluate the limit of $3$ which is constant as $x$ approaches $6$ .

$\sqrt{\frac{{(lim_{x \to 6} x)}^{2} - 1 \cdot 9}{lim_{x \to 6} x - 1 \cdot 3}}$

Step 8

Evaluate the limits by plugging in $6$ for all occurrences of $x$ .

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Step 8.1

Evaluate the limit of $x$ by plugging in $6$ for $x$ .

$\sqrt{\frac{6^{2} - 1 \cdot 9}{lim_{x \to 6} x - 1 \cdot 3}}$

Step 8.2

Evaluate the limit of $x$ by plugging in $6$ for $x$ .

$\sqrt{\frac{6^{2} - 1 \cdot 9}{6 - 1 \cdot 3}}$

Step 9

Simplify the answer.

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Step 9.1

Raise $6$ to the power of $2$ .

$\sqrt{\frac{36 - 1 \cdot 9}{6 - 1 \cdot 3}}$

Step 9.2

Multiply $- 1$ by $9$ .

$\sqrt{\frac{36 - 9}{6 - 1 \cdot 3}}$

Step 9.3

Subtract $9$ from $36$ .

$\sqrt{\frac{27}{6 - 1 \cdot 3}}$

Step 9.4

Multiply $- 1$ by $3$ .

$\sqrt{\frac{27}{6 - 3}}$

Step 9.5

Subtract $3$ from $6$ .

$\sqrt{\frac{27}{3}}$

Step 9.6

Divide $27$ by $3$ .

$\sqrt{9}$

Step 9.7

Rewrite $9$ as $3^{2}$ .

$\sqrt{3^{2}}$

Step 9.8

Pull terms out from under the radical, assuming positive real numbers.

$3$