Calculus Examples

$lim_{x \to - \sqrt{3}} \frac{8 (x^{4} - 9)}{x^{2} - 3}$

Step 1

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$8 lim_{x \to - \sqrt{3}} \frac{x^{4} - 9}{x^{2} - 3}$

Step 2

Apply L'Hospital's rule.

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Step 2.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 2.1.1

Take the limit of the numerator and the limit of the denominator.

$8 \frac{lim_{x \to - \sqrt{3}} x^{4} - 9}{lim_{x \to - \sqrt{3}} x^{2} - 3}$

Step 2.1.2

Evaluate the limit of the numerator.

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Step 2.1.2.1

Evaluate the limit.

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Step 2.1.2.1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- \sqrt{3}$ .

$8 \frac{lim_{x \to - \sqrt{3}} x^{4} - lim_{x \to - \sqrt{3}} 9}{lim_{x \to - \sqrt{3}} x^{2} - 3}$

Step 2.1.2.1.2

Move the exponent $4$ from $x^{4}$ outside the limit using the Limits Power Rule.

$8 \frac{{(lim_{x \to - \sqrt{3}} x)}^{4} - lim_{x \to - \sqrt{3}} 9}{lim_{x \to - \sqrt{3}} x^{2} - 3}$

Step 2.1.2.1.3

Evaluate the limit of $9$ which is constant as $x$ approaches $- \sqrt{3}$ .

$8 \frac{{(lim_{x \to - \sqrt{3}} x)}^{4} - 1 \cdot 9}{lim_{x \to - \sqrt{3}} x^{2} - 3}$

Step 2.1.2.2

Evaluate the limit of $x$ by plugging in $- \sqrt{3}$ for $x$ .

$8 \frac{{(- \sqrt{3})}^{4} - 1 \cdot 9}{lim_{x \to - \sqrt{3}} x^{2} - 3}$

Step 2.1.2.3

Simplify the answer.

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Step 2.1.2.3.1

Simplify each term.

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Step 2.1.2.3.1.1

Apply the product rule to $- \sqrt{3}$ .

$8 \frac{{(- 1)}^{4} {\sqrt{3}}^{4} - 1 \cdot 9}{lim_{x \to - \sqrt{3}} x^{2} - 3}$

Step 2.1.2.3.1.2

Raise $- 1$ to the power of $4$ .

$8 \frac{1 {\sqrt{3}}^{4} - 1 \cdot 9}{lim_{x \to - \sqrt{3}} x^{2} - 3}$

Step 2.1.2.3.1.3

Multiply ${\sqrt{3}}^{4}$ by $1$ .

$8 \frac{{\sqrt{3}}^{4} - 1 \cdot 9}{lim_{x \to - \sqrt{3}} x^{2} - 3}$

Step 2.1.2.3.1.4

Rewrite ${\sqrt{3}}^{4}$ as $3^{2}$ .

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Step 2.1.2.3.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$8 \frac{{(3^{\frac{1}{2}})}^{4} - 1 \cdot 9}{lim_{x \to - \sqrt{3}} x^{2} - 3}$

Step 2.1.2.3.1.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$8 \frac{3^{\frac{1}{2} \cdot 4} - 1 \cdot 9}{lim_{x \to - \sqrt{3}} x^{2} - 3}$

Step 2.1.2.3.1.4.3

Combine $\frac{1}{2}$ and $4$ .

$8 \frac{3^{\frac{4}{2}} - 1 \cdot 9}{lim_{x \to - \sqrt{3}} x^{2} - 3}$

Step 2.1.2.3.1.4.4

Cancel the common factor of $4$ and $2$ .

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Step 2.1.2.3.1.4.4.1

Factor $2$ out of $4$ .

$8 \frac{3^{\frac{2 \cdot 2}{2}} - 1 \cdot 9}{lim_{x \to - \sqrt{3}} x^{2} - 3}$

Step 2.1.2.3.1.4.4.2

Cancel the common factors.

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Step 2.1.2.3.1.4.4.2.1

Factor $2$ out of $2$ .

$8 \frac{3^{\frac{2 \cdot 2}{2 (1)}} - 1 \cdot 9}{lim_{x \to - \sqrt{3}} x^{2} - 3}$

Step 2.1.2.3.1.4.4.2.2

Cancel the common factor.

$8 \frac{3^{\frac{2 \cdot 2}{2 \cdot 1}} - 1 \cdot 9}{lim_{x \to - \sqrt{3}} x^{2} - 3}$

Step 2.1.2.3.1.4.4.2.3

Rewrite the expression.

$8 \frac{3^{\frac{2}{1}} - 1 \cdot 9}{lim_{x \to - \sqrt{3}} x^{2} - 3}$

Step 2.1.2.3.1.4.4.2.4

Divide $2$ by $1$ .

$8 \frac{3^{2} - 1 \cdot 9}{lim_{x \to - \sqrt{3}} x^{2} - 3}$

Step 2.1.2.3.1.5

Raise $3$ to the power of $2$ .

$8 \frac{9 - 1 \cdot 9}{lim_{x \to - \sqrt{3}} x^{2} - 3}$

Step 2.1.2.3.1.6

Multiply $- 1$ by $9$ .

$8 \frac{9 - 9}{lim_{x \to - \sqrt{3}} x^{2} - 3}$

Step 2.1.2.3.2

Subtract $9$ from $9$ .

$8 \frac{0}{lim_{x \to - \sqrt{3}} x^{2} - 3}$

Step 2.1.3

Evaluate the limit of the denominator.

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Step 2.1.3.1

Evaluate the limit.

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Step 2.1.3.1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- \sqrt{3}$ .

$8 \frac{0}{lim_{x \to - \sqrt{3}} x^{2} - lim_{x \to - \sqrt{3}} 3}$

Step 2.1.3.1.2

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

$8 \frac{0}{{(lim_{x \to - \sqrt{3}} x)}^{2} - lim_{x \to - \sqrt{3}} 3}$

Step 2.1.3.1.3

Evaluate the limit of $3$ which is constant as $x$ approaches $- \sqrt{3}$ .

$8 \frac{0}{{(lim_{x \to - \sqrt{3}} x)}^{2} - 1 \cdot 3}$

Step 2.1.3.2

Evaluate the limit of $x$ by plugging in $- \sqrt{3}$ for $x$ .

$8 \frac{0}{{(- \sqrt{3})}^{2} - 1 \cdot 3}$

Step 2.1.3.3

Simplify the answer.

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Step 2.1.3.3.1

Simplify each term.

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Step 2.1.3.3.1.1

Apply the product rule to $- \sqrt{3}$ .

$8 \frac{0}{{(- 1)}^{2} {\sqrt{3}}^{2} - 1 \cdot 3}$

Step 2.1.3.3.1.2

Raise $- 1$ to the power of $2$ .

$8 \frac{0}{1 {\sqrt{3}}^{2} - 1 \cdot 3}$

Step 2.1.3.3.1.3

Multiply ${\sqrt{3}}^{2}$ by $1$ .

$8 \frac{0}{{\sqrt{3}}^{2} - 1 \cdot 3}$

Step 2.1.3.3.1.4

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.1.3.3.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$8 \frac{0}{{(3^{\frac{1}{2}})}^{2} - 1 \cdot 3}$

Step 2.1.3.3.1.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$8 \frac{0}{3^{\frac{1}{2} \cdot 2} - 1 \cdot 3}$

Step 2.1.3.3.1.4.3

Combine $\frac{1}{2}$ and $2$ .

$8 \frac{0}{3^{\frac{2}{2}} - 1 \cdot 3}$

Step 2.1.3.3.1.4.4

Cancel the common factor of $2$ .

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Step 2.1.3.3.1.4.4.1

Cancel the common factor.

$8 \frac{0}{3^{\frac{2}{2}} - 1 \cdot 3}$

Step 2.1.3.3.1.4.4.2

Rewrite the expression.

$8 \frac{0}{3^{1} - 1 \cdot 3}$

Step 2.1.3.3.1.4.5

Evaluate the exponent.

$8 \frac{0}{3 - 1 \cdot 3}$

Step 2.1.3.3.1.5

Multiply $- 1$ by $3$ .

$8 \frac{0}{3 - 3}$

Step 2.1.3.3.2

Subtract $3$ from $3$ .

$8 (\frac{0}{0})$

Step 2.1.3.3.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

$8 (\frac{0}{0})$

Step 2.1.3.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$8 (\frac{0}{0})$

Step 2.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$8 (\frac{0}{0})$

Step 2.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to - \sqrt{3}} \frac{x^{4} - 9}{x^{2} - 3} = lim_{x \to - \sqrt{3}} \frac{\frac{d}{d x} [x^{4} - 9]}{\frac{d}{d x} [x^{2} - 3]}$

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

$8 lim_{x \to - \sqrt{3}} \frac{\frac{d}{d x} [x^{4} - 9]}{\frac{d}{d x} [x^{2} - 3]}$

Step 2.3.2

By the Sum Rule, the derivative of $x^{4} - 9$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 9]$ .

$8 lim_{x \to - \sqrt{3}} \frac{\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 9]}{\frac{d}{d x} [x^{2} - 3]}$

Step 2.3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$8 lim_{x \to - \sqrt{3}} \frac{4 x^{3} + \frac{d}{d x} [- 9]}{\frac{d}{d x} [x^{2} - 3]}$

Step 2.3.4

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9$ with respect to $x$ is $0$ .

$8 lim_{x \to - \sqrt{3}} \frac{4 x^{3} + 0}{\frac{d}{d x} [x^{2} - 3]}$

Step 2.3.5

Add $4 x^{3}$ and $0$ .

$8 lim_{x \to - \sqrt{3}} \frac{4 x^{3}}{\frac{d}{d x} [x^{2} - 3]}$

Step 2.3.6

By the Sum Rule, the derivative of $x^{2} - 3$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3]$ .

$8 lim_{x \to - \sqrt{3}} \frac{4 x^{3}}{\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3]}$

Step 2.3.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$8 lim_{x \to - \sqrt{3}} \frac{4 x^{3}}{2 x + \frac{d}{d x} [- 3]}$

Step 2.3.8

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$8 lim_{x \to - \sqrt{3}} \frac{4 x^{3}}{2 x + 0}$

Step 2.3.9

Add $2 x$ and $0$ .

$8 lim_{x \to - \sqrt{3}} \frac{4 x^{3}}{2 x}$

Step 2.4

Reduce.

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Step 2.4.1

Cancel the common factor of $4$ and $2$ .

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Step 2.4.1.1

Factor $2$ out of $4 x^{3}$ .

$8 lim_{x \to - \sqrt{3}} \frac{2 \cdot 2 x^{3}}{2 x}$

Step 2.4.1.2

Cancel the common factors.

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Step 2.4.1.2.1

Factor $2$ out of $2 x$ .

$8 lim_{x \to - \sqrt{3}} \frac{2 \cdot 2 x^{3}}{2 (x)}$

Step 2.4.1.2.2

Cancel the common factor.

$8 lim_{x \to - \sqrt{3}} \frac{2 \cdot 2 x^{3}}{2 x}$

Step 2.4.1.2.3

Rewrite the expression.

$8 lim_{x \to - \sqrt{3}} \frac{2 x^{3}}{x}$

Step 2.4.2

Cancel the common factor of $x^{3}$ and $x$ .

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Step 2.4.2.1

Factor $x$ out of $2 x^{3}$ .

$8 lim_{x \to - \sqrt{3}} \frac{x \cdot 2 x^{2}}{x}$

Step 2.4.2.2

Cancel the common factors.

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Step 2.4.2.2.1

Raise $x$ to the power of $1$ .

$8 lim_{x \to - \sqrt{3}} \frac{x \cdot 2 x^{2}}{x^{1}}$

Step 2.4.2.2.2

Factor $x$ out of $x^{1}$ .

$8 lim_{x \to - \sqrt{3}} \frac{x \cdot 2 x^{2}}{x \cdot 1}$

Step 2.4.2.2.3

Cancel the common factor.

$8 lim_{x \to - \sqrt{3}} \frac{x \cdot 2 x^{2}}{x \cdot 1}$

Step 2.4.2.2.4

Rewrite the expression.

$8 lim_{x \to - \sqrt{3}} \frac{2 x^{2}}{1}$

Step 2.4.2.2.5

Divide $2 x^{2}$ by $1$ .

$8 lim_{x \to - \sqrt{3}} 2 x^{2}$

Step 3

Evaluate the limit.

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Step 3.1

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

$8 \cdot 2 lim_{x \to - \sqrt{3}} x^{2}$

Step 3.2

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

$8 \cdot 2 {(lim_{x \to - \sqrt{3}} x)}^{2}$

Step 4

Evaluate the limit of $x$ by plugging in $- \sqrt{3}$ for $x$ .

$8 \cdot 2 {(- \sqrt{3})}^{2}$

Step 5

Simplify the answer.

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Step 5.1

Multiply $8$ by $2$ .

$16 {(- \sqrt{3})}^{2}$

Step 5.2

Apply the product rule to $- \sqrt{3}$ .

$16 ({(- 1)}^{2} {\sqrt{3}}^{2})$

Step 5.3

Raise $- 1$ to the power of $2$ .

$16 (1 {\sqrt{3}}^{2})$

Step 5.4

Multiply ${\sqrt{3}}^{2}$ by $1$ .

$16 {\sqrt{3}}^{2}$

Step 5.5

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 5.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$16 {(3^{\frac{1}{2}})}^{2}$

Step 5.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$16 \cdot 3^{\frac{1}{2} \cdot 2}$

Step 5.5.3

Combine $\frac{1}{2}$ and $2$ .

$16 \cdot 3^{\frac{2}{2}}$

Step 5.5.4

Cancel the common factor of $2$ .

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Step 5.5.4.1

Cancel the common factor.

$16 \cdot 3^{\frac{2}{2}}$

Step 5.5.4.2

Rewrite the expression.

$16 \cdot 3^{1}$

Step 5.5.5

Evaluate the exponent.

$16 \cdot 3$

Step 5.6

Multiply $16$ by $3$ .

$48$