Calculus Examples

$\frac{lim_{x \to 4} 4 - x}{5 - \sqrt{x^{2} - 9}}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$\frac{lim_{x \to 4} 4 - lim_{x \to 4} x}{5 - \sqrt{x^{2} - 9}}$

Step 2

Evaluate the limit of $4$ which is constant as $x$ approaches $4$ .

$\frac{4 - lim_{x \to 4} x}{5 - \sqrt{x^{2} - 9}}$

Step 3

Simplify terms.

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Step 3.1

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\frac{4 - 4}{5 - \sqrt{x^{2} - 9}}$

Step 3.2

Simplify the answer.

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Step 3.2.1

Subtract $4$ from $4$ .

$\frac{0}{5 - \sqrt{x^{2} - 9}}$

Step 3.2.2

Simplify the denominator.

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Step 3.2.2.1

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

$\frac{0}{5 - \sqrt{x^{2} - 3^{2}}}$

Step 3.2.2.2

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 $b = 3$ .

$\frac{0}{5 - \sqrt{(x + 3) (x - 3)}}$

Step 3.2.3

Divide $0$ by $5 - \sqrt{(x + 3) (x - 3)}$ .

$0$