Calculus Examples

$\frac{lim_{x \to 0} 6 x - 9}{x^{3} - 12 x + 3}$

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 $0$ .

$\frac{lim_{x \to 0} 6 x - lim_{x \to 0} 9}{x^{3} - 12 x + 3}$

Step 1.2

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\frac{6 lim_{x \to 0} x - lim_{x \to 0} 9}{x^{3} - 12 x + 3}$

Step 1.3

Evaluate the limit of $9$ which is constant as $x$ approaches $0$ .

$\frac{6 lim_{x \to 0} x - 1 \cdot 9}{x^{3} - 12 x + 3}$

Step 2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{6 \cdot 0 - 1 \cdot 9}{x^{3} - 12 x + 3}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Multiply $6$ by $0$ .

$\frac{0 - 1 \cdot 9}{x^{3} - 12 x + 3}$

Step 3.1.2

Multiply $- 1$ by $9$ .

$\frac{0 - 9}{x^{3} - 12 x + 3}$

Step 3.1.3

Subtract $9$ from $0$ .

$\frac{- 9}{x^{3} - 12 x + 3}$

Step 3.2

Move the negative in front of the fraction.

$- \frac{9}{x^{3} - 12 x + 3}$