Calculus Examples

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

Step 1

Move the term $\frac{1}{4}$ outside of the limit because it is constant with respect to $x$ .

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Evaluate the limit of $3$ which is constant as $x$ approaches $- 1$ .

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

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 $x$ .

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

Step 6.2

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

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

Step 6.3

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

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

Step 7

Simplify the answer.

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Step 7.1

Move the negative one from the denominator of $\frac{- {(- 1)}^{2} - 1 + 3}{- 1}$ .

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

Step 7.2

Rewrite $- 1 \cdot (- {(- 1)}^{2} - 1 + 3)$ as $- (- {(- 1)}^{2} - 1 + 3)$ .

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

Step 7.3

Simplify each term.

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Step 7.3.1

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 7.3.1.1

Multiply $- 1$ by ${(- 1)}^{2}$ .

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Step 7.3.1.1.1

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

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

Step 7.3.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 7.3.1.2

Add $1$ and $2$ .

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

Step 7.3.2

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

$\frac{1}{4} (- (- 1 - 1 + 3))$

Step 7.4

Subtract $1$ from $- 1$ .

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

Step 7.5

Add $- 2$ and $3$ .

$\frac{1}{4} (- 1 \cdot 1)$

Step 7.6

Multiply $- 1$ by $1$ .

$\frac{1}{4} \cdot - 1$

Step 7.7

Combine $\frac{1}{4}$ and $- 1$ .

$\frac{- 1}{4}$

Step 7.8

Move the negative in front of the fraction.

$- \frac{1}{4}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{4}$

Decimal Form:

$- 0.25$