Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Move the term $9$ outside of the limit because it is constant with respect to $x$ .

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

Step 4

Evaluate the limits by plugging in $3$ for all occurrences of $x$ .

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Step 4.1

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

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

Step 4.2

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

$\frac{3^{3} - 9 \cdot 3}{x - 3}$

Step 5

Simplify the answer.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

$\frac{27 - 9 \cdot 3}{x - 3}$

Step 5.1.2

Multiply $- 9$ by $3$ .

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

Step 5.1.3

Subtract $27$ from $27$ .

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

Step 5.2

Divide $0$ by $x - 3$ .

$0$