Calculus Examples

$\frac{lim_{x \to 4} \sqrt{x^{2} - 3 x} - 2}{5 - \sqrt{x^{2} + 9}}$

Step 1

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

$\frac{lim_{x \to 4} \sqrt{x^{2} - 3 x} - lim_{x \to 4} 2}{5 - \sqrt{x^{2} + 9}}$

Step 2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 4} x^{2} - 3 x} - lim_{x \to 4} 2}{5 - \sqrt{x^{2} + 9}}$

Step 3

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

$\frac{\sqrt{lim_{x \to 4} x^{2} - lim_{x \to 4} 3 x} - lim_{x \to 4} 2}{5 - \sqrt{x^{2} + 9}}$

Step 4

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

$\frac{\sqrt{{(lim_{x \to 4} x)}^{2} - lim_{x \to 4} 3 x} - lim_{x \to 4} 2}{5 - \sqrt{x^{2} + 9}}$

Step 5

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sqrt{{(lim_{x \to 4} x)}^{2} - 3 lim_{x \to 4} x} - lim_{x \to 4} 2}{5 - \sqrt{x^{2} + 9}}$

Step 6

Evaluate the limit of $2$ which is constant as $x$ approaches $4$ .

$\frac{\sqrt{{(lim_{x \to 4} x)}^{2} - 3 lim_{x \to 4} x} - 1 \cdot 2}{5 - \sqrt{x^{2} + 9}}$

Step 7

Evaluate the limits by plugging in $4$ for all occurrences of $x$ .

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Step 7.1

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

$\frac{\sqrt{4^{2} - 3 lim_{x \to 4} x} - 1 \cdot 2}{5 - \sqrt{x^{2} + 9}}$

Step 7.2

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

$\frac{\sqrt{4^{2} - 3 \cdot 4} - 1 \cdot 2}{5 - \sqrt{x^{2} + 9}}$

Step 8

Simplify the answer.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $4$ to the power of $2$ .

$\frac{\sqrt{16 - 3 \cdot 4} - 1 \cdot 2}{5 - \sqrt{x^{2} + 9}}$

Step 8.1.2

Multiply $- 3$ by $4$ .

$\frac{\sqrt{16 - 12} - 1 \cdot 2}{5 - \sqrt{x^{2} + 9}}$

Step 8.1.3

Subtract $12$ from $16$ .

$\frac{\sqrt{4} - 1 \cdot 2}{5 - \sqrt{x^{2} + 9}}$

Step 8.1.4

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt{2^{2}} - 1 \cdot 2}{5 - \sqrt{x^{2} + 9}}$

Step 8.1.5

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

$\frac{2 - 1 \cdot 2}{5 - \sqrt{x^{2} + 9}}$

Step 8.1.6

Multiply $- 1$ by $2$ .

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

Step 8.1.7

Subtract $2$ from $2$ .

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

Step 8.2

Divide $0$ by $5 - \sqrt{x^{2} + 9}$ .

$0$