Calculus Examples

$lim_{x \to - 2} {(3 x^{4} + 7 x^{3} + x^{2} - 6 x - 10)}^{8}$

Step 1

Move the exponent $8$ from ${(3 x^{4} + 7 x^{3} + x^{2} - 6 x - 10)}^{8}$ outside the limit using the Limits Power Rule.

${(lim_{x \to - 2} 3 x^{4} + 7 x^{3} + x^{2} - 6 x - 10)}^{8}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 2$ .

${(lim_{x \to - 2} 3 x^{4} + lim_{x \to - 2} 7 x^{3} + lim_{x \to - 2} x^{2} - lim_{x \to - 2} 6 x - lim_{x \to - 2} 10)}^{8}$

Step 3

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

${(3 lim_{x \to - 2} x^{4} + lim_{x \to - 2} 7 x^{3} + lim_{x \to - 2} x^{2} - lim_{x \to - 2} 6 x - lim_{x \to - 2} 10)}^{8}$

Step 4

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

${(3 {(lim_{x \to - 2} x)}^{4} + lim_{x \to - 2} 7 x^{3} + lim_{x \to - 2} x^{2} - lim_{x \to - 2} 6 x - lim_{x \to - 2} 10)}^{8}$

Step 5

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

${(3 {(lim_{x \to - 2} x)}^{4} + 7 lim_{x \to - 2} x^{3} + lim_{x \to - 2} x^{2} - lim_{x \to - 2} 6 x - lim_{x \to - 2} 10)}^{8}$

Step 6

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

${(3 {(lim_{x \to - 2} x)}^{4} + 7 {(lim_{x \to - 2} x)}^{3} + lim_{x \to - 2} x^{2} - lim_{x \to - 2} 6 x - lim_{x \to - 2} 10)}^{8}$

Step 7

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

${(3 {(lim_{x \to - 2} x)}^{4} + 7 {(lim_{x \to - 2} x)}^{3} + {(lim_{x \to - 2} x)}^{2} - lim_{x \to - 2} 6 x - lim_{x \to - 2} 10)}^{8}$

Step 8

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

${(3 {(lim_{x \to - 2} x)}^{4} + 7 {(lim_{x \to - 2} x)}^{3} + {(lim_{x \to - 2} x)}^{2} - 6 lim_{x \to - 2} x - lim_{x \to - 2} 10)}^{8}$

Step 9

Evaluate the limit of $10$ which is constant as $x$ approaches $- 2$ .

${(3 {(lim_{x \to - 2} x)}^{4} + 7 {(lim_{x \to - 2} x)}^{3} + {(lim_{x \to - 2} x)}^{2} - 6 lim_{x \to - 2} x - 1 \cdot 10)}^{8}$

Step 10

Evaluate the limits by plugging in $- 2$ for all occurrences of $x$ .

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Step 10.1

Evaluate the limit of $x$ by plugging in $- 2$ for $x$ .

${(3 {(- 2)}^{4} + 7 {(lim_{x \to - 2} x)}^{3} + {(lim_{x \to - 2} x)}^{2} - 6 lim_{x \to - 2} x - 1 \cdot 10)}^{8}$

Step 10.2

Evaluate the limit of $x$ by plugging in $- 2$ for $x$ .

${(3 {(- 2)}^{4} + 7 {(- 2)}^{3} + {(lim_{x \to - 2} x)}^{2} - 6 lim_{x \to - 2} x - 1 \cdot 10)}^{8}$

Step 10.3

Evaluate the limit of $x$ by plugging in $- 2$ for $x$ .

${(3 {(- 2)}^{4} + 7 {(- 2)}^{3} + {(- 2)}^{2} - 6 lim_{x \to - 2} x - 1 \cdot 10)}^{8}$

Step 10.4

Evaluate the limit of $x$ by plugging in $- 2$ for $x$ .

${(3 {(- 2)}^{4} + 7 {(- 2)}^{3} + {(- 2)}^{2} - 6 \cdot - 2 - 1 \cdot 10)}^{8}$

Step 11

Simplify the answer.

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Step 11.1

Simplify each term.

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Step 11.1.1

Raise $- 2$ to the power of $4$ .

${(3 \cdot 16 + 7 {(- 2)}^{3} + {(- 2)}^{2} - 6 \cdot - 2 - 1 \cdot 10)}^{8}$

Step 11.1.2

Multiply $3$ by $16$ .

${(48 + 7 {(- 2)}^{3} + {(- 2)}^{2} - 6 \cdot - 2 - 1 \cdot 10)}^{8}$

Step 11.1.3

Raise $- 2$ to the power of $3$ .

${(48 + 7 \cdot - 8 + {(- 2)}^{2} - 6 \cdot - 2 - 1 \cdot 10)}^{8}$

Step 11.1.4

Multiply $7$ by $- 8$ .

${(48 - 56 + {(- 2)}^{2} - 6 \cdot - 2 - 1 \cdot 10)}^{8}$

Step 11.1.5

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

${(48 - 56 + 4 - 6 \cdot - 2 - 1 \cdot 10)}^{8}$

Step 11.1.6

Multiply $- 6$ by $- 2$ .

${(48 - 56 + 4 + 12 - 1 \cdot 10)}^{8}$

Step 11.1.7

Multiply $- 1$ by $10$ .

${(48 - 56 + 4 + 12 - 10)}^{8}$

Step 11.2

Subtract $56$ from $48$ .

${(- 8 + 4 + 12 - 10)}^{8}$

Step 11.3

Add $- 8$ and $4$ .

${(- 4 + 12 - 10)}^{8}$

Step 11.4

Add $- 4$ and $12$ .

${(8 - 10)}^{8}$

Step 11.5

Subtract $10$ from $8$ .

${(- 2)}^{8}$

Step 11.6

Raise $- 2$ to the power of $8$ .

$256$