Calculus Examples

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

Step 1

Apply L'Hospital's rule.

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Evaluate the limit.

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Step 1.1.2.1.1

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

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

Step 1.1.2.1.2

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

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

Step 1.1.2.1.3

Evaluate the limit of $8$ which is constant as $x$ approaches $2$ .

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

Step 1.1.2.2

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

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

Step 1.1.2.3

Simplify the answer.

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Step 1.1.2.3.1

Simplify each term.

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Step 1.1.2.3.1.1

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

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

Step 1.1.2.3.1.2

Multiply $- 1$ by $8$ .

$\frac{8 - 8}{lim_{x \to 2} x - 2}$

Step 1.1.2.3.2

Subtract $8$ from $8$ .

$\frac{0}{lim_{x \to 2} x - 2}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

Evaluate the limit.

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Step 1.1.3.1.1

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

$\frac{0}{lim_{x \to 2} x - lim_{x \to 2} 2}$

Step 1.1.3.1.2

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

$\frac{0}{lim_{x \to 2} x - 1 \cdot 2}$

Step 1.1.3.2

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

$\frac{0}{2 - 1 \cdot 2}$

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Multiply $- 1$ by $2$ .

$\frac{0}{2 - 2}$

Step 1.1.3.3.2

Subtract $2$ from $2$ .

$\frac{0}{0}$

Step 1.1.3.3.3

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

Undefined

$\frac{0}{0}$

Step 1.1.3.4

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

Undefined

$\frac{0}{0}$

Step 1.1.4

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

Undefined

$\frac{0}{0}$

Step 1.2

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.

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

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

By the Sum Rule, the derivative of $x^{3} - 8$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 8]$ .

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

Step 1.3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

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

Step 1.3.4

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8$ with respect to $x$ is $0$ .

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

Step 1.3.5

Add $3 x^{2}$ and $0$ .

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

Step 1.3.6

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

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

Step 1.3.7

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

$lim_{x \to 2} \frac{3 x^{2}}{1 + \frac{d}{d x} [- 2]}$

Step 1.3.8

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$lim_{x \to 2} \frac{3 x^{2}}{1 + 0}$

Step 1.3.9

Add $1$ and $0$ .

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

Step 1.4

Divide $3 x^{2}$ by $1$ .

$lim_{x \to 2} 3 x^{2}$

Step 2

Evaluate the limit.

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Step 2.1

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

$3 lim_{x \to 2} x^{2}$

Step 2.2

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

$3 {(lim_{x \to 2} x)}^{2}$

Step 3

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

$3 \cdot 2^{2}$

Step 4

Simplify the answer.

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Step 4.1

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

$3 \cdot 4$

Step 4.2

Multiply $3$ by $4$ .

$12$