Calculus Examples

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

Step 1

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

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

Step 2

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

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

Step 3

Move the exponent $4$ from $x^{4}$ outside the limit using the Limits Power Rule.

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

Step 4

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

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

Step 5

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

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

Step 6

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

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

Step 7

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

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

Step 8

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

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

Step 9

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

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Step 9.1

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

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

Step 9.2

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

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

Step 9.3

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

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

Step 10

Simplify the answer.

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Step 10.1

Simplify each term.

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Step 10.1.1

Raise $- 1$ to the power of $4$ .

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

Step 10.1.2

Multiply $2$ by $1$ .

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

Step 10.1.3

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

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

Step 10.1.4

Multiply $- 2$ by $- 1$ .

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

Step 10.1.5

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

$2 + 2 - 2 \cdot 1 - 1 \cdot 2$

Step 10.1.6

Multiply $- 2$ by $1$ .

$2 + 2 - 2 - 1 \cdot 2$

Step 10.1.7

Multiply $- 1$ by $2$ .

$2 + 2 - 2 - 2$

Step 10.2

Add $2$ and $2$ .

$4 - 2 - 2$

Step 10.3

Subtract $2$ from $4$ .

$2 - 2$

Step 10.4

Subtract $2$ from $2$ .

$0$