Calculus Examples

$lim_{x \to - 2} x^{3} - 6 x + 1$

Step 1

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

$lim_{x \to - 2} x^{3} - lim_{x \to - 2} 6 x + lim_{x \to - 2} 1$

Step 2

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

${(lim_{x \to - 2} x)}^{3} - lim_{x \to - 2} 6 x + lim_{x \to - 2} 1$

Step 3

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

${(lim_{x \to - 2} x)}^{3} - 6 lim_{x \to - 2} x + lim_{x \to - 2} 1$

Step 4

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

${(lim_{x \to - 2} x)}^{3} - 6 lim_{x \to - 2} x + 1$

Step 5

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

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Step 5.1

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

${(- 2)}^{3} - 6 lim_{x \to - 2} x + 1$

Step 5.2

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

${(- 2)}^{3} - 6 \cdot - 2 + 1$

Step 6

Simplify the answer.

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Step 6.1

Simplify each term.

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Step 6.1.1

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

$- 8 - 6 \cdot - 2 + 1$

Step 6.1.2

Multiply $- 6$ by $- 2$ .

$- 8 + 12 + 1$

Step 6.2

Add $- 8$ and $12$ .

$4 + 1$

Step 6.3

Add $4$ and $1$ .

$5$