Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

Simplify each term.

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Step 3.1.1

One to any power is one.

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

Step 3.1.2

Multiply $- 1$ by $2$ .

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

Step 3.2

Subtract $2$ from $1$ .

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

Step 3.3

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

$- 1$