Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 7

Simplify the answer.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

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

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

Step 7.1.2

Cancel the common factor of $3$ .

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Step 7.1.2.1

Factor $3$ out of $3^{2}$ .

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

Step 7.1.2.2

Cancel the common factor.

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

Step 7.1.2.3

Rewrite the expression.

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

Step 7.1.3

One to any power is one.

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

Step 7.1.4

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

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

Step 7.1.5

Multiply $- 1$ by $1$ .

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

Step 7.1.6

Combine the numerators over the common denominator.

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

Step 7.1.7

Add $1$ and $2$ .

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

Step 7.1.8

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 7.1.9

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

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

Step 7.1.10

Combine the numerators over the common denominator.

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

Step 7.1.11

Simplify the numerator.

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Step 7.1.11.1

Multiply $- 1$ by $3$ .

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

Step 7.1.11.2

Subtract $3$ from $3$ .

$\frac{\frac{0}{3}}{9 x^{2} - 1}$

Step 7.1.12

Divide $0$ by $3$ .

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

Step 7.2

Simplify the denominator.

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Step 7.2.1

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

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

Step 7.2.2

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

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

Step 7.2.3

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 7.3

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

$0$