Calculus Examples

$lim_{x \to - 1} {(x + x^{2} + x^{3})}^{135}$

Step 1

Move the exponent $135$ from ${(x + x^{2} + x^{3})}^{135}$ outside the limit using the Limits Power Rule.

${(lim_{x \to - 1} x + x^{2} + x^{3})}^{135}$

Step 2

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

${(lim_{x \to - 1} x + lim_{x \to - 1} x^{2} + lim_{x \to - 1} x^{3})}^{135}$

Step 3

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

${(lim_{x \to - 1} x + {(lim_{x \to - 1} x)}^{2} + lim_{x \to - 1} x^{3})}^{135}$

Step 4

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

${(lim_{x \to - 1} x + {(lim_{x \to - 1} x)}^{2} + {(lim_{x \to - 1} x)}^{3})}^{135}$

Step 5

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

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Step 5.1

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

${(- 1 + {(lim_{x \to - 1} x)}^{2} + {(lim_{x \to - 1} x)}^{3})}^{135}$

Step 5.2

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

${(- 1 + {(- 1)}^{2} + {(lim_{x \to - 1} x)}^{3})}^{135}$

Step 5.3

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

${(- 1 + {(- 1)}^{2} + {(- 1)}^{3})}^{135}$

Step 6

Simplify the answer.

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Step 6.1

Simplify each term.

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Step 6.1.1

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

${(- 1 + 1 + {(- 1)}^{3})}^{135}$

Step 6.1.2

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

${(- 1 + 1 - 1)}^{135}$

Step 6.2

Add $- 1$ and $1$ .

${(0 - 1)}^{135}$

Step 6.3

Subtract $1$ from $0$ .

${(- 1)}^{135}$

Step 6.4

Raise $- 1$ to the power of $135$ .

$- 1$