Calculus Examples

$lim_{x \to 1} - (7 x^{2} - 5 x - 4) (x^{\frac{1}{3}} + 8)$

Step 1

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

$- lim_{x \to 1} (7 x^{2} - 5 x - 4) (x^{\frac{1}{3}} + 8)$

Step 2

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

$- lim_{x \to 1} 7 x^{2} - 5 x - 4 \cdot lim_{x \to 1} x^{\frac{1}{3}} + 8$

Step 3

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

$- (lim_{x \to 1} 7 x^{2} - lim_{x \to 1} 5 x - lim_{x \to 1} 4) \cdot lim_{x \to 1} x^{\frac{1}{3}} + 8$

Step 4

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$- (7 lim_{x \to 1} x^{2} - lim_{x \to 1} 5 x - lim_{x \to 1} 4) \cdot lim_{x \to 1} x^{\frac{1}{3}} + 8$

Step 5

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

$- (7 {(lim_{x \to 1} x)}^{2} - lim_{x \to 1} 5 x - lim_{x \to 1} 4) \cdot lim_{x \to 1} x^{\frac{1}{3}} + 8$

Step 6

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

$- (7 {(lim_{x \to 1} x)}^{2} - 5 lim_{x \to 1} x - lim_{x \to 1} 4) \cdot lim_{x \to 1} x^{\frac{1}{3}} + 8$

Step 7

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

$- (7 {(lim_{x \to 1} x)}^{2} - 5 lim_{x \to 1} x - 1 \cdot 4) \cdot lim_{x \to 1} x^{\frac{1}{3}} + 8$

Step 8

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

$- (7 {(lim_{x \to 1} x)}^{2} - 5 lim_{x \to 1} x - 1 \cdot 4) \cdot (lim_{x \to 1} x^{\frac{1}{3}} + lim_{x \to 1} 8)$

Step 9

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

$- (7 {(lim_{x \to 1} x)}^{2} - 5 lim_{x \to 1} x - 1 \cdot 4) \cdot ({(lim_{x \to 1} x)}^{\frac{1}{3}} + lim_{x \to 1} 8)$

Step 10

Evaluate the limit of $8$ which is constant as $x$ approaches $1$ .

$- (7 {(lim_{x \to 1} x)}^{2} - 5 lim_{x \to 1} x - 1 \cdot 4) \cdot ({(lim_{x \to 1} x)}^{\frac{1}{3}} + 8)$

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 $x$ .

$- (7 \cdot 1^{2} - 5 lim_{x \to 1} x - 1 \cdot 4) \cdot ({(lim_{x \to 1} x)}^{\frac{1}{3}} + 8)$

Step 11.2

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

$- (7 \cdot 1^{2} - 5 \cdot 1 - 1 \cdot 4) \cdot ({(lim_{x \to 1} x)}^{\frac{1}{3}} + 8)$

Step 11.3

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

$- (7 \cdot 1^{2} - 5 \cdot 1 - 1 \cdot 4) \cdot (1^{\frac{1}{3}} + 8)$

Step 12

Simplify the answer.

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Step 12.1

Simplify each term.

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Step 12.1.1

One to any power is one.

$- (7 \cdot 1 - 5 \cdot 1 - 1 \cdot 4) \cdot (1^{\frac{1}{3}} + 8)$

Step 12.1.2

Multiply $7$ by $1$ .

$- (7 - 5 \cdot 1 - 1 \cdot 4) \cdot (1^{\frac{1}{3}} + 8)$

Step 12.1.3

Multiply $- 5$ by $1$ .

$- (7 - 5 - 1 \cdot 4) \cdot (1^{\frac{1}{3}} + 8)$

Step 12.1.4

Multiply $- 1$ by $4$ .

$- (7 - 5 - 4) \cdot (1^{\frac{1}{3}} + 8)$

Step 12.2

Subtract $5$ from $7$ .

$- (2 - 4) \cdot (1^{\frac{1}{3}} + 8)$

Step 12.3

Subtract $4$ from $2$ .

$- - 2 \cdot (1^{\frac{1}{3}} + 8)$

Step 12.4

Multiply $- 1$ by $- 2$ .

$2 \cdot (1^{\frac{1}{3}} + 8)$

Step 12.5

One to any power is one.

$2 \cdot (1 + 8)$

Step 12.6

Add $1$ and $8$ .

$2 \cdot 9$

Step 12.7

Multiply $2$ by $9$ .

$18$