Calculus Examples

$\frac{lim_{x \to 6} x^{3} - 36 x}{x - 6}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $6$ .

$\frac{lim_{x \to 6} x^{3} - lim_{x \to 6} 36 x}{x - 6}$

Step 2

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

$\frac{{(lim_{x \to 6} x)}^{3} - lim_{x \to 6} 36 x}{x - 6}$

Step 3

Move the term $36$ outside of the limit because it is constant with respect to $x$ .

$\frac{{(lim_{x \to 6} x)}^{3} - 36 lim_{x \to 6} x}{x - 6}$

Step 4

Evaluate the limits by plugging in $6$ for all occurrences of $x$ .

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Step 4.1

Evaluate the limit of $x$ by plugging in $6$ for $x$ .

$\frac{6^{3} - 36 lim_{x \to 6} x}{x - 6}$

Step 4.2

Evaluate the limit of $x$ by plugging in $6$ for $x$ .

$\frac{6^{3} - 36 \cdot 6}{x - 6}$

Step 5

Simplify the answer.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $6$ to the power of $3$ .

$\frac{216 - 36 \cdot 6}{x - 6}$

Step 5.1.2

Multiply $- 36$ by $6$ .

$\frac{216 - 216}{x - 6}$

Step 5.1.3

Subtract $216$ from $216$ .

$\frac{0}{x - 6}$

Step 5.2

Divide $0$ by $x - 6$ .

$0$