Calculus Examples

$lim_{x \to - 1} - (9 x^{2} - 5 x - 4) (x^{\frac{1}{3}} - 2)$

Step 1

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

$- lim_{x \to - 1} (9 x^{2} - 5 x - 4) (x^{\frac{1}{3}} - 2)$

Step 2

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $- 1$ .

$- lim_{x \to - 1} 9 x^{2} - 5 x - 4 \cdot lim_{x \to - 1} x^{\frac{1}{3}} - 2$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 1$ .

$- (lim_{x \to - 1} 9 x^{2} - lim_{x \to - 1} 5 x - lim_{x \to - 1} 4) \cdot lim_{x \to - 1} x^{\frac{1}{3}} - 2$

Step 4

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

$- (9 lim_{x \to - 1} x^{2} - lim_{x \to - 1} 5 x - lim_{x \to - 1} 4) \cdot lim_{x \to - 1} x^{\frac{1}{3}} - 2$

Step 5

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

$- (9 {(lim_{x \to - 1} x)}^{2} - lim_{x \to - 1} 5 x - lim_{x \to - 1} 4) \cdot lim_{x \to - 1} x^{\frac{1}{3}} - 2$

Step 6

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$- (9 {(lim_{x \to - 1} x)}^{2} - 5 lim_{x \to - 1} x - lim_{x \to - 1} 4) \cdot lim_{x \to - 1} x^{\frac{1}{3}} - 2$

Step 7

Evaluate the limit of $4$ which is constant as $x$ approaches $- 1$ .

$- (9 {(lim_{x \to - 1} x)}^{2} - 5 lim_{x \to - 1} x - 1 \cdot 4) \cdot lim_{x \to - 1} x^{\frac{1}{3}} - 2$

Step 8

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 1$ .

$- (9 {(lim_{x \to - 1} x)}^{2} - 5 lim_{x \to - 1} x - 1 \cdot 4) \cdot (lim_{x \to - 1} x^{\frac{1}{3}} - lim_{x \to - 1} 2)$

Step 9

Move the exponent $\frac{1}{3}$ from $x^{\frac{1}{3}}$ outside the limit using the Limits Power Rule.

$- (9 {(lim_{x \to - 1} x)}^{2} - 5 lim_{x \to - 1} x - 1 \cdot 4) \cdot ({(lim_{x \to - 1} x)}^{\frac{1}{3}} - lim_{x \to - 1} 2)$

Step 10

Evaluate the limit of $2$ which is constant as $x$ approaches $- 1$ .

$- (9 {(lim_{x \to - 1} x)}^{2} - 5 lim_{x \to - 1} x - 1 \cdot 4) \cdot ({(lim_{x \to - 1} x)}^{\frac{1}{3}} - 1 \cdot 2)$

Step 11

Evaluate the limits by plugging in $- 1$ for all occurrences of $x$ .

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Step 11.1

Evaluate the limit of $x$ by plugging in $- 1$ for $x$ .

$- (9 {(- 1)}^{2} - 5 lim_{x \to - 1} x - 1 \cdot 4) \cdot ({(lim_{x \to - 1} x)}^{\frac{1}{3}} - 1 \cdot 2)$

Step 11.2

Evaluate the limit of $x$ by plugging in $- 1$ for $x$ .

$- (9 {(- 1)}^{2} - 5 \cdot - 1 - 1 \cdot 4) \cdot ({(lim_{x \to - 1} x)}^{\frac{1}{3}} - 1 \cdot 2)$

Step 11.3

Evaluate the limit of $x$ by plugging in $- 1$ for $x$ .

$- (9 {(- 1)}^{2} - 5 \cdot - 1 - 1 \cdot 4) \cdot ({(- 1)}^{\frac{1}{3}} - 1 \cdot 2)$

Step 12

Simplify the answer.

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Step 12.1

Simplify each term.

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Step 12.1.1

Raise $- 1$ to the power of $2$ .

$- (9 \cdot 1 - 5 \cdot - 1 - 1 \cdot 4) \cdot ({(- 1)}^{\frac{1}{3}} - 1 \cdot 2)$

Step 12.1.2

Multiply $9$ by $1$ .

$- (9 - 5 \cdot - 1 - 1 \cdot 4) \cdot ({(- 1)}^{\frac{1}{3}} - 1 \cdot 2)$

Step 12.1.3

Multiply $- 5$ by $- 1$ .

$- (9 + 5 - 1 \cdot 4) \cdot ({(- 1)}^{\frac{1}{3}} - 1 \cdot 2)$

Step 12.1.4

Multiply $- 1$ by $4$ .

$- (9 + 5 - 4) \cdot ({(- 1)}^{\frac{1}{3}} - 1 \cdot 2)$

Step 12.2

Add $9$ and $5$ .

$- (14 - 4) \cdot ({(- 1)}^{\frac{1}{3}} - 1 \cdot 2)$

Step 12.3

Subtract $4$ from $14$ .

$- 1 \cdot 10 \cdot ({(- 1)}^{\frac{1}{3}} - 1 \cdot 2)$

Step 12.4

Multiply $- 1$ by $10$ .

$- 10 \cdot ({(- 1)}^{\frac{1}{3}} - 1 \cdot 2)$

Step 12.5

Simplify each term.

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Step 12.5.1

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$- 10 \cdot ({({(- 1)}^{3})}^{\frac{1}{3}} - 1 \cdot 2)$

Step 12.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- 10 \cdot ({(- 1)}^{3 (\frac{1}{3})} - 1 \cdot 2)$

Step 12.5.3

Cancel the common factor of $3$ .

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Step 12.5.3.1

Cancel the common factor.

$- 10 \cdot ({(- 1)}^{3 (\frac{1}{3})} - 1 \cdot 2)$

Step 12.5.3.2

Rewrite the expression.

$- 10 \cdot ({(- 1)}^{1} - 1 \cdot 2)$

Step 12.5.4

Evaluate the exponent.

$- 10 \cdot (- 1 - 1 \cdot 2)$

Step 12.5.5

Multiply $- 1$ by $2$ .

$- 10 \cdot (- 1 - 2)$

Step 12.6

Subtract $2$ from $- 1$ .

$- 10 \cdot - 3$

Step 12.7

Multiply $- 10$ by $- 3$ .

$30$