Precalculus Examples

$lim_{x \to 0} \frac{5 (3 x + 6)}{5} \cdot ((x + 6) (- 6 x^{2} - 20 x - 16))$

Step 1

Cancel the common factor of $5$ .

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Step 1.1

Cancel the common factor.

$lim_{x \to 0} \frac{5 (3 x + 6)}{5} \cdot ((x + 6) (- 6 x^{2} - 20 x - 16))$

Step 1.2

Divide $3 x + 6$ by $1$ .

$lim_{x \to 0} (3 x + 6) \cdot ((x + 6) (- 6 x^{2} - 20 x - 16))$

Step 2

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

$lim_{x \to 0} 3 x + 6 \cdot lim_{x \to 0} x + 6 \cdot lim_{x \to 0} - 6 x^{2} - 20 x - 16$

Step 3

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

$(lim_{x \to 0} 3 x + lim_{x \to 0} 6) \cdot lim_{x \to 0} x + 6 \cdot lim_{x \to 0} - 6 x^{2} - 20 x - 16$

Step 4

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$(3 lim_{x \to 0} x + lim_{x \to 0} 6) \cdot lim_{x \to 0} x + 6 \cdot lim_{x \to 0} - 6 x^{2} - 20 x - 16$

Step 5

Evaluate the limit of $6$ which is constant as $x$ approaches $0$ .

$(3 lim_{x \to 0} x + 6) \cdot lim_{x \to 0} x + 6 \cdot lim_{x \to 0} - 6 x^{2} - 20 x - 16$

Step 6

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

$(3 lim_{x \to 0} x + 6) \cdot (lim_{x \to 0} x + lim_{x \to 0} 6) \cdot lim_{x \to 0} - 6 x^{2} - 20 x - 16$

Step 7

Evaluate the limit of $6$ which is constant as $x$ approaches $0$ .

$(3 lim_{x \to 0} x + 6) \cdot (lim_{x \to 0} x + 6) \cdot lim_{x \to 0} - 6 x^{2} - 20 x - 16$

Step 8

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

$(3 lim_{x \to 0} x + 6) \cdot (lim_{x \to 0} x + 6) \cdot (- lim_{x \to 0} 6 x^{2} - lim_{x \to 0} 20 x - lim_{x \to 0} 16)$

Step 9

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

$(3 lim_{x \to 0} x + 6) \cdot (lim_{x \to 0} x + 6) \cdot (- 6 lim_{x \to 0} x^{2} - lim_{x \to 0} 20 x - lim_{x \to 0} 16)$

Step 10

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$(3 lim_{x \to 0} x + 6) \cdot (lim_{x \to 0} x + 6) \cdot (- 6 {(lim_{x \to 0} x)}^{2} - lim_{x \to 0} 20 x - lim_{x \to 0} 16)$

Step 11

Move the term $20$ outside of the limit because it is constant with respect to $x$ .

$(3 lim_{x \to 0} x + 6) \cdot (lim_{x \to 0} x + 6) \cdot (- 6 {(lim_{x \to 0} x)}^{2} - 20 lim_{x \to 0} x - lim_{x \to 0} 16)$

Step 12

Evaluate the limit of $16$ which is constant as $x$ approaches $0$ .

$(3 lim_{x \to 0} x + 6) \cdot (lim_{x \to 0} x + 6) \cdot (- 6 {(lim_{x \to 0} x)}^{2} - 20 lim_{x \to 0} x - 1 \cdot 16)$

Step 13

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

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Step 13.1

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

$(3 \cdot 0 + 6) \cdot (lim_{x \to 0} x + 6) \cdot (- 6 {(lim_{x \to 0} x)}^{2} - 20 lim_{x \to 0} x - 1 \cdot 16)$

Step 13.2

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

$(3 \cdot 0 + 6) \cdot (0 + 6) \cdot (- 6 {(lim_{x \to 0} x)}^{2} - 20 lim_{x \to 0} x - 1 \cdot 16)$

Step 13.3

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

$(3 \cdot 0 + 6) \cdot (0 + 6) \cdot (- 6 \cdot 0^{2} - 20 lim_{x \to 0} x - 1 \cdot 16)$

Step 13.4

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

$(3 \cdot 0 + 6) \cdot (0 + 6) \cdot (- 6 \cdot 0^{2} - 20 \cdot 0 - 1 \cdot 16)$

Step 14

Simplify the answer.

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Step 14.1

Multiply $3$ by $0$ .

$(0 + 6) \cdot (0 + 6) \cdot (- 6 \cdot 0^{2} - 20 \cdot 0 - 1 \cdot 16)$

Step 14.2

Add $0$ and $6$ .

$6 \cdot (0 + 6) \cdot (- 6 \cdot 0^{2} - 20 \cdot 0 - 1 \cdot 16)$

Step 14.3

Add $0$ and $6$ .

$6 \cdot 6 \cdot (- 6 \cdot 0^{2} - 20 \cdot 0 - 1 \cdot 16)$

Step 14.4

Multiply $6$ by $6$ .

$36 \cdot (- 6 \cdot 0^{2} - 20 \cdot 0 - 1 \cdot 16)$

Step 14.5

Simplify each term.

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Step 14.5.1

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

$36 \cdot (- 6 \cdot 0 - 20 \cdot 0 - 1 \cdot 16)$

Step 14.5.2

Multiply $- 6$ by $0$ .

$36 \cdot (0 - 20 \cdot 0 - 1 \cdot 16)$

Step 14.5.3

Multiply $- 20$ by $0$ .

$36 \cdot (0 + 0 - 1 \cdot 16)$

Step 14.5.4

Multiply $- 1$ by $16$ .

$36 \cdot (0 + 0 - 16)$

Step 14.6

Add $0$ and $0$ .

$36 \cdot (0 - 16)$

Step 14.7

Subtract $16$ from $0$ .

$36 \cdot - 16$

Step 14.8

Multiply $36$ by $- 16$ .

$- 576$