Write the Fraction in Simplest Form limit as x approaches 2 of (x^3-8)/(x-2)

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

Step 1

Apply L\'Hospital\'s rule.

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

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

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

Evaluate the limit of the numerator.

Evaluate the limit.

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

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

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

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

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

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

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

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

Simplify the answer.

Simplify each term.

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

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

Multiply $- 1$ by $8$ .

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

Subtract $8$ from $8$ .

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

Evaluate the limit of the denominator.

Evaluate the limit.

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

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

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

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

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

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

Simplify the answer.

Multiply $- 1$ by $2$ .

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

Subtract $2$ from $2$ .

$\frac{0}{0}$

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

Undefined

$\frac{0}{0}$

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

Undefined

$\frac{0}{0}$

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

Undefined

$\frac{0}{0}$

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.

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

Find the derivative of the numerator and denominator.

Differentiate the numerator and denominator.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Add $1$ and $0$ .

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

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

$3 \cdot 2^{2}$

Step 5

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

$3 \cdot 4$

Step 6

Multiply $3$ by $4$ .

$12$

Write the Fraction in Simplest Form limit as x approaches 2 of (x^3-8)/(x-2)

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