Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

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

Step 6

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

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

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 $x$ .

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

Step 7.2

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

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

Step 8

Simplify the answer.

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Step 8.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 8.1.1

Apply the product rule to $- \frac{1}{2}$ .

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

Step 8.1.2

Apply the product rule to $\frac{1}{2}$ .

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

Step 8.2

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

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

Step 8.3

Multiply $\frac{1^{2}}{2^{2}}$ by $1$ .

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

Step 8.4

One to any power is one.

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

Step 8.5

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

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

Step 8.6

Combine $3$ and $\frac{1}{4}$ .

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

Step 8.7

Simplify each term.

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Step 8.7.1

Cancel the common factor of $2$ .

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Step 8.7.1.1

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

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

Step 8.7.1.2

Cancel the common factor.

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

Step 8.7.1.3

Rewrite the expression.

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

Step 8.7.2

Multiply $- 1$ by $1$ .

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

Step 8.8

Subtract $1$ from $- 1$ .

$\frac{3}{4} \cdot - 2$

Step 8.9

Cancel the common factor of $2$ .

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Step 8.9.1

Factor $2$ out of $4$ .

$\frac{3}{2 (2)} \cdot - 2$

Step 8.9.2

Factor $2$ out of $- 2$ .

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

Step 8.9.3

Cancel the common factor.

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

Step 8.9.4

Rewrite the expression.

$\frac{3}{2} \cdot - 1$

Step 8.10

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

$\frac{3 \cdot - 1}{2}$

Step 8.11

Multiply $3$ by $- 1$ .

$\frac{- 3}{2}$

Step 8.12

Move the negative in front of the fraction.

$- \frac{3}{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \frac{3}{2}$

Decimal Form:

$- 1.5$