Calculus Examples

$\frac{lim_{x \to 2} \sqrt[3]{x^{2} + 4} - \sqrt{x^{2} + 6 x}}{x^{2} - 4}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{lim_{x \to 2} \sqrt[3]{x^{2} + 4} - lim_{x \to 2} \sqrt{x^{2} + 6 x}}{x^{2} - 4}$

Step 2

Move the limit under the radical sign.

$\frac{\sqrt[3]{lim_{x \to 2} x^{2} + 4} - lim_{x \to 2} \sqrt{x^{2} + 6 x}}{x^{2} - 4}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{\sqrt[3]{lim_{x \to 2} x^{2} + lim_{x \to 2} 4} - lim_{x \to 2} \sqrt{x^{2} + 6 x}}{x^{2} - 4}$

Step 4

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

$\frac{\sqrt[3]{{(lim_{x \to 2} x)}^{2} + lim_{x \to 2} 4} - lim_{x \to 2} \sqrt{x^{2} + 6 x}}{x^{2} - 4}$

Step 5

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

$\frac{\sqrt[3]{{(lim_{x \to 2} x)}^{2} + 4} - lim_{x \to 2} \sqrt{x^{2} + 6 x}}{x^{2} - 4}$

Step 6

Move the limit under the radical sign.

$\frac{\sqrt[3]{{(lim_{x \to 2} x)}^{2} + 4} - \sqrt{lim_{x \to 2} x^{2} + 6 x}}{x^{2} - 4}$

Step 7

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{\sqrt[3]{{(lim_{x \to 2} x)}^{2} + 4} - \sqrt{lim_{x \to 2} x^{2} + lim_{x \to 2} 6 x}}{x^{2} - 4}$

Step 8

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

$\frac{\sqrt[3]{{(lim_{x \to 2} x)}^{2} + 4} - \sqrt{{(lim_{x \to 2} x)}^{2} + lim_{x \to 2} 6 x}}{x^{2} - 4}$

Step 9

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sqrt[3]{{(lim_{x \to 2} x)}^{2} + 4} - \sqrt{{(lim_{x \to 2} x)}^{2} + 6 lim_{x \to 2} x}}{x^{2} - 4}$

Step 10

Evaluate the limits by plugging in $2$ for all occurrences of $x$ .

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Step 10.1

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{\sqrt[3]{2^{2} + 4} - \sqrt{{(lim_{x \to 2} x)}^{2} + 6 lim_{x \to 2} x}}{x^{2} - 4}$

Step 10.2

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{\sqrt[3]{2^{2} + 4} - \sqrt{2^{2} + 6 lim_{x \to 2} x}}{x^{2} - 4}$

Step 10.3

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{\sqrt[3]{2^{2} + 4} - \sqrt{2^{2} + 6 \cdot 2}}{x^{2} - 4}$

Step 11

Simplify the answer.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

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

$\frac{\sqrt[3]{4 + 4} - \sqrt{2^{2} + 6 \cdot 2}}{x^{2} - 4}$

Step 11.1.2

Add $4$ and $4$ .

$\frac{\sqrt[3]{8} - \sqrt{2^{2} + 6 \cdot 2}}{x^{2} - 4}$

Step 11.1.3

Rewrite $8$ as $2^{3}$ .

$\frac{\sqrt[3]{2^{3}} - \sqrt{2^{2} + 6 \cdot 2}}{x^{2} - 4}$

Step 11.1.4

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

$\frac{2 - \sqrt{2^{2} + 6 \cdot 2}}{x^{2} - 4}$

Step 11.1.5

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

$\frac{2 - \sqrt{4 + 6 \cdot 2}}{x^{2} - 4}$

Step 11.1.6

Multiply $6$ by $2$ .

$\frac{2 - \sqrt{4 + 12}}{x^{2} - 4}$

Step 11.1.7

Add $4$ and $12$ .

$\frac{2 - \sqrt{16}}{x^{2} - 4}$

Step 11.1.8

Rewrite $16$ as $4^{2}$ .

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

Step 11.1.9

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

$\frac{2 - 1 \cdot 4}{x^{2} - 4}$

Step 11.1.10

Multiply $- 1$ by $4$ .

$\frac{2 - 4}{x^{2} - 4}$

Step 11.1.11

Subtract $4$ from $2$ .

$\frac{- 2}{x^{2} - 4}$

Step 11.2

Simplify the denominator.

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Step 11.2.1

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

$\frac{- 2}{x^{2} - 2^{2}}$

Step 11.2.2

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 $b = 2$ .

$\frac{- 2}{(x + 2) (x - 2)}$

Step 11.3

Move the negative in front of the fraction.

$- \frac{2}{(x + 2) (x - 2)}$