Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

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

Step 10

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

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

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 $x$ .

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

Step 11.2

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

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

Step 11.3

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

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

Step 12

Simplify the answer.

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Step 12.1

Simplify the numerator.

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Step 12.1.1

Raise $- 1$ to the power of $3$ .

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

Step 12.1.2

Multiply $- 5$ by $- 1$ .

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

Step 12.1.3

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

$\frac{5 + 5 \cdot 1 + 2}{4 \cdot - 1 - 1 \cdot 2}$

Step 12.1.4

Multiply $5$ by $1$ .

$\frac{5 + 5 + 2}{4 \cdot - 1 - 1 \cdot 2}$

Step 12.1.5

Add $5$ and $5$ .

$\frac{10 + 2}{4 \cdot - 1 - 1 \cdot 2}$

Step 12.1.6

Add $10$ and $2$ .

$\frac{12}{4 \cdot - 1 - 1 \cdot 2}$

Step 12.2

Simplify the denominator.

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Step 12.2.1

Multiply $4$ by $- 1$ .

$\frac{12}{- 4 - 1 \cdot 2}$

Step 12.2.2

Multiply $- 1$ by $2$ .

$\frac{12}{- 4 - 2}$

Step 12.2.3

Subtract $2$ from $- 4$ .

$\frac{12}{- 6}$

Step 12.3

Divide $12$ by $- 6$ .

$- 2$