Calculus Examples

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

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 $\frac{1}{2}$ .

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

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

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

Step 3.1.2

One to any power is one.

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

Step 3.1.3

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

$\frac{\frac{1}{4} - 1 \cdot 1}{2 x + 1}$

Step 3.1.4

Multiply $- 1$ by $1$ .

$\frac{\frac{1}{4} - 1}{2 x + 1}$

Step 3.1.5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{\frac{1}{4} - 1 \cdot \frac{4}{4}}{2 x + 1}$

Step 3.1.6

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

$\frac{\frac{1}{4} + \frac{- 1 \cdot 4}{4}}{2 x + 1}$

Step 3.1.7

Combine the numerators over the common denominator.

$\frac{\frac{1 - 1 \cdot 4}{4}}{2 x + 1}$

Step 3.1.8

Simplify the numerator.

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Step 3.1.8.1

Multiply $- 1$ by $4$ .

$\frac{\frac{1 - 4}{4}}{2 x + 1}$

Step 3.1.8.2

Subtract $4$ from $1$ .

$\frac{\frac{- 3}{4}}{2 x + 1}$

Step 3.1.9

Move the negative in front of the fraction.

$\frac{- \frac{3}{4}}{2 x + 1}$

Step 3.2

Multiply the numerator by the reciprocal of the denominator.

$- \frac{3}{4} \cdot \frac{1}{2 x + 1}$

Step 3.3

Multiply $\frac{1}{2 x + 1}$ by $\frac{3}{4}$ .

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

Step 3.4

Move $4$ to the left of $2 x + 1$ .

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