Calculus Examples

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

Step 1

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

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

Step 2

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $- \frac{1}{2}$ .

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

Step 3

Move the exponent $2$ from ${(3 x + 4)}^{2}$ outside the limit using the Limits Power Rule.

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

Step 4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- \frac{1}{2}$ .

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

Step 5

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

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

Step 6

Evaluate the limit of $4$ which is constant as $x$ approaches $- \frac{1}{2}$ .

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

Step 7

Evaluate the limits by plugging in $- \frac{1}{2}$ for all occurrences of $x$ .

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Step 7.1

Evaluate the limit of $x$ by plugging in $- \frac{1}{2}$ for $x$ .

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

Step 7.2

Evaluate the limit of $x$ by plugging in $- \frac{1}{2}$ for $x$ .

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

Step 8

Simplify the answer.

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Step 8.1

Cancel the common factor of $2$ .

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Step 8.1.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

Step 8.1.2

Factor $2$ out of $4$ .

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

Step 8.1.3

Cancel the common factor.

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

Step 8.1.4

Rewrite the expression.

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

Step 8.2

Multiply $2$ by $- 1$ .

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

Step 8.3

Simplify each term.

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Step 8.3.1

Multiply $3 (- \frac{1}{2})$ .

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Step 8.3.1.1

Multiply $- 1$ by $3$ .

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

Step 8.3.1.2

Combine $- 3$ and $\frac{1}{2}$ .

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

Step 8.3.2

Move the negative in front of the fraction.

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

Step 8.4

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 8.5

Combine $4$ and $\frac{2}{2}$ .

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

Step 8.6

Combine the numerators over the common denominator.

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

Step 8.7

Simplify the numerator.

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Step 8.7.1

Multiply $4$ by $2$ .

$- 2 \cdot {(\frac{- 3 + 8}{2})}^{2}$

Step 8.7.2

Add $- 3$ and $8$ .

$- 2 \cdot {(\frac{5}{2})}^{2}$

Step 8.8

Apply the product rule to $\frac{5}{2}$ .

$- 2 \cdot \frac{5^{2}}{2^{2}}$

Step 8.9

Raise $5$ to the power of $2$ .

$- 2 \cdot \frac{25}{2^{2}}$

Step 8.10

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

$- 2 \cdot \frac{25}{4}$

Step 8.11

Cancel the common factor of $2$ .

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Step 8.11.1

Factor $2$ out of $- 2$ .

$2 (- 1) \cdot \frac{25}{4}$

Step 8.11.2

Factor $2$ out of $4$ .

$2 \cdot - 1 \cdot \frac{25}{2 \cdot 2}$

Step 8.11.3

Cancel the common factor.

$2 \cdot - 1 \cdot \frac{25}{2 \cdot 2}$

Step 8.11.4

Rewrite the expression.

$- 1 \cdot \frac{25}{2}$

Step 8.12

Rewrite $- 1 (\frac{25}{2})$ as $- (\frac{25}{2})$ .

$- \frac{25}{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \frac{25}{2}$

Decimal Form:

$- 12.5$