Calculus Examples

$lim_{x \to - \frac{1}{3}} (x^{2} + \frac{1}{9}) (x - \frac{1}{3})$

Step 1

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $- \frac{1}{3}$ .

$lim_{x \to - \frac{1}{3}} x^{2} + \frac{1}{9} \cdot lim_{x \to - \frac{1}{3}} x - \frac{1}{3}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- \frac{1}{3}$ .

$(lim_{x \to - \frac{1}{3}} x^{2} + lim_{x \to - \frac{1}{3}} \frac{1}{9}) \cdot lim_{x \to - \frac{1}{3}} x - \frac{1}{3}$

Step 3

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

$({(lim_{x \to - \frac{1}{3}} x)}^{2} + lim_{x \to - \frac{1}{3}} \frac{1}{9}) \cdot lim_{x \to - \frac{1}{3}} x - \frac{1}{3}$

Step 4

Evaluate the limit of $\frac{1}{9}$ which is constant as $x$ approaches $- \frac{1}{3}$ .

$({(lim_{x \to - \frac{1}{3}} x)}^{2} + \frac{1}{9}) \cdot lim_{x \to - \frac{1}{3}} x - \frac{1}{3}$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- \frac{1}{3}$ .

$({(lim_{x \to - \frac{1}{3}} x)}^{2} + \frac{1}{9}) \cdot (lim_{x \to - \frac{1}{3}} x - lim_{x \to - \frac{1}{3}} \frac{1}{3})$

Step 6

Evaluate the limit of $\frac{1}{3}$ which is constant as $x$ approaches $- \frac{1}{3}$ .

$({(lim_{x \to - \frac{1}{3}} x)}^{2} + \frac{1}{9}) \cdot (lim_{x \to - \frac{1}{3}} x - \frac{1}{3})$

Step 7

Evaluate the limits by plugging in $- \frac{1}{3}$ for all occurrences of $x$ .

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Step 7.1

Evaluate the limit of $x$ by plugging in $- \frac{1}{3}$ for $x$ .

$({(- \frac{1}{3})}^{2} + \frac{1}{9}) \cdot (lim_{x \to - \frac{1}{3}} x - \frac{1}{3})$

Step 7.2

Evaluate the limit of $x$ by plugging in $- \frac{1}{3}$ for $x$ .

$({(- \frac{1}{3})}^{2} + \frac{1}{9}) \cdot (- \frac{1}{3} - \frac{1}{3})$

Step 8

Simplify the answer.

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Step 8.1

Simplify each term.

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Step 8.1.1

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

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Step 8.1.1.1

Apply the product rule to $- \frac{1}{3}$ .

$({(- 1)}^{2} {(\frac{1}{3})}^{2} + \frac{1}{9}) \cdot (- \frac{1}{3} - \frac{1}{3})$

Step 8.1.1.2

Apply the product rule to $\frac{1}{3}$ .

$({(- 1)}^{2} \frac{1^{2}}{3^{2}} + \frac{1}{9}) \cdot (- \frac{1}{3} - \frac{1}{3})$

Step 8.1.2

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

$(1 \frac{1^{2}}{3^{2}} + \frac{1}{9}) \cdot (- \frac{1}{3} - \frac{1}{3})$

Step 8.1.3

Multiply $\frac{1^{2}}{3^{2}}$ by $1$ .

$(\frac{1^{2}}{3^{2}} + \frac{1}{9}) \cdot (- \frac{1}{3} - \frac{1}{3})$

Step 8.1.4

One to any power is one.

$(\frac{1}{3^{2}} + \frac{1}{9}) \cdot (- \frac{1}{3} - \frac{1}{3})$

Step 8.1.5

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

$(\frac{1}{9} + \frac{1}{9}) \cdot (- \frac{1}{3} - \frac{1}{3})$

Step 8.2

Combine the numerators over the common denominator.

$\frac{1 + 1}{9} \cdot (- \frac{1}{3} - \frac{1}{3})$

Step 8.3

Add $1$ and $1$ .

$\frac{2}{9} \cdot (- \frac{1}{3} - \frac{1}{3})$

Step 8.4

Combine the numerators over the common denominator.

$\frac{2}{9} \cdot \frac{- 1 - 1}{3}$

Step 8.5

Subtract $1$ from $- 1$ .

$\frac{2}{9} \cdot \frac{- 2}{3}$

Step 8.6

Move the negative in front of the fraction.

$\frac{2}{9} \cdot (- \frac{2}{3})$

Step 8.7

Multiply $\frac{2}{9} (- \frac{2}{3})$ .

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Step 8.7.1

Multiply $\frac{2}{9}$ by $\frac{2}{3}$ .

$- \frac{2 \cdot 2}{9 \cdot 3}$

Step 8.7.2

Multiply $2$ by $2$ .

$- \frac{4}{9 \cdot 3}$

Step 8.7.3

Multiply $9$ by $3$ .

$- \frac{4}{27}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \frac{4}{27}$

Decimal Form:

$- 0.\overline{148}$