Calculus Examples

$\frac{lim_{x \to - 1} x^{3} - 4 x^{2} + x + 6}{x + 1}$

Step 1

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

$\frac{lim_{x \to - 1} x^{3} - lim_{x \to - 1} 4 x^{2} + lim_{x \to - 1} x + lim_{x \to - 1} 6}{x + 1}$

Step 2

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

$\frac{{(lim_{x \to - 1} x)}^{3} - lim_{x \to - 1} 4 x^{2} + lim_{x \to - 1} x + lim_{x \to - 1} 6}{x + 1}$

Step 3

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{{(lim_{x \to - 1} x)}^{3} - 4 lim_{x \to - 1} x^{2} + lim_{x \to - 1} x + lim_{x \to - 1} 6}{x + 1}$

Step 4

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

$\frac{{(lim_{x \to - 1} x)}^{3} - 4 {(lim_{x \to - 1} x)}^{2} + lim_{x \to - 1} x + lim_{x \to - 1} 6}{x + 1}$

Step 5

Evaluate the limit of $6$ which is constant as $x$ approaches $- 1$ .

$\frac{{(lim_{x \to - 1} x)}^{3} - 4 {(lim_{x \to - 1} x)}^{2} + lim_{x \to - 1} x + 6}{x + 1}$

Step 6

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

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Step 6.1

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

$\frac{{(- 1)}^{3} - 4 {(lim_{x \to - 1} x)}^{2} + lim_{x \to - 1} x + 6}{x + 1}$

Step 6.2

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

$\frac{{(- 1)}^{3} - 4 {(- 1)}^{2} + lim_{x \to - 1} x + 6}{x + 1}$

Step 6.3

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

$\frac{{(- 1)}^{3} - 4 {(- 1)}^{2} - 1 + 6}{x + 1}$

Step 7

Simplify the answer.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

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

$\frac{- 1 - 4 {(- 1)}^{2} - 1 + 6}{x + 1}$

Step 7.1.2

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

$\frac{- 1 - 4 \cdot 1 - 1 + 6}{x + 1}$

Step 7.1.3

Multiply $- 4$ by $1$ .

$\frac{- 1 - 4 - 1 + 6}{x + 1}$

Step 7.1.4

Subtract $4$ from $- 1$ .

$\frac{- 5 - 1 + 6}{x + 1}$

Step 7.1.5

Subtract $1$ from $- 5$ .

$\frac{- 6 + 6}{x + 1}$

Step 7.1.6

Add $- 6$ and $6$ .

$\frac{0}{x + 1}$

Step 7.2

Divide $0$ by $x + 1$ .

$0$