Calculus Examples

$\frac{lim_{x \to 3} x^{3} - 27}{x^{2} - 9}$

Step 1

Evaluate the limit.

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Step 1.1

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

$\frac{lim_{x \to 3} x^{3} - lim_{x \to 3} 27}{x^{2} - 9}$

Step 1.2

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

$\frac{{(lim_{x \to 3} x)}^{3} - lim_{x \to 3} 27}{x^{2} - 9}$

Step 1.3

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

$\frac{{(lim_{x \to 3} x)}^{3} - 1 \cdot 27}{x^{2} - 9}$

Step 2

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

$\frac{3^{3} - 1 \cdot 27}{x^{2} - 9}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

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

$\frac{27 - 1 \cdot 27}{x^{2} - 9}$

Step 3.1.2

Multiply $- 1$ by $27$ .

$\frac{27 - 27}{x^{2} - 9}$

Step 3.1.3

Subtract $27$ from $27$ .

$\frac{0}{x^{2} - 9}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Rewrite $9$ as $3^{2}$ .

$\frac{0}{x^{2} - 3^{2}}$

Step 3.2.2

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 $b = 3$ .

$\frac{0}{(x + 3) (x - 3)}$

Step 3.3

Divide $0$ by $(x + 3) (x - 3)$ .

$0$