Write the Fraction in Simplest Form limit as x approaches 3 of (x^3-27)/(x-3)

$lim_{x \to 3} \frac{x^{3} - 27}{x - 3}$

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.

$\frac{lim_{x \to 3} x^{3} - 27}{lim_{x \to 3} x - 3}$

Evaluate the limit of the numerator.

Evaluate the limit.

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

$\frac{lim_{x \to 3} x^{3} - lim_{x \to 3} 27}{lim_{x \to 3} x - 3}$

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

$\frac{{(lim_{x \to 3} x)}^{3} - lim_{x \to 3} 27}{lim_{x \to 3} x - 3}$

Evaluate the limit of $27$ which is constant as $x$ approaches $3$ .

$\frac{{(lim_{x \to 3} x)}^{3} - 1 \cdot 27}{lim_{x \to 3} x - 3}$

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{3^{3} - 1 \cdot 27}{lim_{x \to 3} x - 3}$

Simplify the answer.

Simplify each term.

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

$\frac{27 - 1 \cdot 27}{lim_{x \to 3} x - 3}$

Multiply $- 1$ by $27$ .

$\frac{27 - 27}{lim_{x \to 3} x - 3}$

Subtract $27$ from $27$ .

$\frac{0}{lim_{x \to 3} x - 3}$

Evaluate the limit of the denominator.

Evaluate the limit.

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

$\frac{0}{lim_{x \to 3} x - lim_{x \to 3} 3}$

Evaluate the limit of $3$ which is constant as $x$ approaches $3$ .

$\frac{0}{lim_{x \to 3} x - 1 \cdot 3}$

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

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

Simplify the answer.

Multiply $- 1$ by $3$ .

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

Subtract $3$ from $3$ .

$\frac{0}{0}$

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.

$lim_{x \to 3} \frac{x^{3} - 27}{x - 3} = lim_{x \to 3} \frac{\frac{d}{d x} [x^{3} - 27]}{\frac{d}{d x} [x - 3]}$

Find the derivative of the numerator and denominator.

Differentiate the numerator and denominator.

$lim_{x \to 3} \frac{\frac{d}{d x} [x^{3} - 27]}{\frac{d}{d x} [x - 3]}$

By the Sum Rule, the derivative of $x^{3} - 27$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 27]$ .

$lim_{x \to 3} \frac{\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 27]}{\frac{d}{d x} [x - 3]}$

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

$lim_{x \to 3} \frac{3 x^{2} + \frac{d}{d x} [- 27]}{\frac{d}{d x} [x - 3]}$

Since $- 27$ is constant with respect to $x$ , the derivative of $- 27$ with respect to $x$ is $0$ .

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

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

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

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

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

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

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

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

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

Add $1$ and $0$ .

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

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

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

Step 2

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

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

Step 3

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

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

Step 4

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$3 \cdot 3^{2}$

Step 5

Multiply $3$ by $3^{2}$ by adding the exponents.

Multiply $3$ by $3^{2}$ .

Raise $3$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$3^{1 + 2}$

Add $1$ and $2$ .

$3^{3}$

Step 6

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

$27$

Write the Fraction in Simplest Form limit as x approaches 3 of (x^3-27)/(x-3)

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