Calculus Examples

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

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $0$ .

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

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

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

Step 6

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

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

Step 7

Move the term $12$ outside of the limit because it is constant with respect to $x$ .

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

Step 8

Evaluate the limit of $3$ which is constant as $x$ approaches $0$ .

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

Step 9

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

Tap for more steps...

Step 9.1

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

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

Step 9.2

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

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

Step 9.3

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

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

Step 10

Simplify the answer.

Tap for more steps...

Step 10.1

Simplify the numerator.

Tap for more steps...

Step 10.1.1

Multiply $6$ by $0$ .

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

Step 10.1.2

Multiply $- 1$ by $9$ .

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

Step 10.1.3

Subtract $9$ from $0$ .

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

Step 10.2

Simplify the denominator.

Tap for more steps...

Step 10.2.1

Raising $0$ to any positive power yields $0$ .

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

Step 10.2.2

Multiply $- 12$ by $0$ .

$\frac{- 9}{0 + 0 + 3}$

Step 10.2.3

Add $0$ and $0$ .

$\frac{- 9}{0 + 3}$

Step 10.2.4

Add $0$ and $3$ .

$\frac{- 9}{3}$

Step 10.3

Divide $- 9$ by $3$ .

$- 3$