Precalculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Cancel the common factor of $3$ .

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Step 3.1.1.1

Cancel the common factor.

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

Step 3.1.1.2

Rewrite the expression.

$\frac{1 - 1 \cdot 1}{9 x^{2} - 1}$

Step 3.1.2

Multiply $- 1$ by $1$ .

$\frac{1 - 1}{9 x^{2} - 1}$

Step 3.1.3

Subtract $1$ from $1$ .

$\frac{0}{9 x^{2} - 1}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Rewrite $9 x^{2}$ as ${(3 x)}^{2}$ .

$\frac{0}{{(3 x)}^{2} - 1}$

Step 3.2.2

Rewrite $1$ as $1^{2}$ .

$\frac{0}{{(3 x)}^{2} - 1^{2}}$

Step 3.2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3 x$ and $b = 1$ .

$\frac{0}{(3 x + 1) (3 x - 1)}$

Step 3.3

Divide $0$ by $(3 x + 1) (3 x - 1)$ .

$0$