Calculus Examples

$\frac{lim_{x \to - \frac{1}{8}} 4 x^{2} + x + \frac{1}{16}}{{(8 x + 1)}^{2}}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- \frac{1}{8}$ .

$\frac{lim_{x \to - \frac{1}{8}} 4 x^{2} + lim_{x \to - \frac{1}{8}} x + lim_{x \to - \frac{1}{8}} \frac{1}{16}}{{(8 x + 1)}^{2}}$

Step 2

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{4 lim_{x \to - \frac{1}{8}} x^{2} + lim_{x \to - \frac{1}{8}} x + lim_{x \to - \frac{1}{8}} \frac{1}{16}}{{(8 x + 1)}^{2}}$

Step 3

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

$\frac{4 {(lim_{x \to - \frac{1}{8}} x)}^{2} + lim_{x \to - \frac{1}{8}} x + lim_{x \to - \frac{1}{8}} \frac{1}{16}}{{(8 x + 1)}^{2}}$

Step 4

Evaluate the limit of $\frac{1}{16}$ which is constant as $x$ approaches $- \frac{1}{8}$ .

$\frac{4 {(lim_{x \to - \frac{1}{8}} x)}^{2} + lim_{x \to - \frac{1}{8}} x + \frac{1}{16}}{{(8 x + 1)}^{2}}$

Step 5

Evaluate the limits by plugging in $- \frac{1}{8}$ for all occurrences of $x$ .

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Step 5.1

Evaluate the limit of $x$ by plugging in $- \frac{1}{8}$ for $x$ .

$\frac{4 {(- \frac{1}{8})}^{2} + lim_{x \to - \frac{1}{8}} x + \frac{1}{16}}{{(8 x + 1)}^{2}}$

Step 5.2

Evaluate the limit of $x$ by plugging in $- \frac{1}{8}$ for $x$ .

$\frac{4 {(- \frac{1}{8})}^{2} - \frac{1}{8} + \frac{1}{16}}{{(8 x + 1)}^{2}}$

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

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Step 6.1.1.1

Apply the product rule to $- \frac{1}{8}$ .

$\frac{4 ({(- 1)}^{2} {(\frac{1}{8})}^{2}) - \frac{1}{8} + \frac{1}{16}}{{(8 x + 1)}^{2}}$

Step 6.1.1.2

Apply the product rule to $\frac{1}{8}$ .

$\frac{4 ({(- 1)}^{2} \frac{1^{2}}{8^{2}}) - \frac{1}{8} + \frac{1}{16}}{{(8 x + 1)}^{2}}$

Step 6.1.2

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

$\frac{4 (1 \frac{1^{2}}{8^{2}}) - \frac{1}{8} + \frac{1}{16}}{{(8 x + 1)}^{2}}$

Step 6.1.3

Multiply $\frac{1^{2}}{8^{2}}$ by $1$ .

$\frac{4 \frac{1^{2}}{8^{2}} - \frac{1}{8} + \frac{1}{16}}{{(8 x + 1)}^{2}}$

Step 6.1.4

One to any power is one.

$\frac{4 \frac{1}{8^{2}} - \frac{1}{8} + \frac{1}{16}}{{(8 x + 1)}^{2}}$

Step 6.1.5

Raise $8$ to the power of $2$ .

$\frac{4 (\frac{1}{64}) - \frac{1}{8} + \frac{1}{16}}{{(8 x + 1)}^{2}}$

Step 6.1.6

Cancel the common factor of $4$ .

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Step 6.1.6.1

Factor $4$ out of $64$ .

$\frac{4 \frac{1}{4 (16)} - \frac{1}{8} + \frac{1}{16}}{{(8 x + 1)}^{2}}$

Step 6.1.6.2

Cancel the common factor.

$\frac{4 \frac{1}{4 \cdot 16} - \frac{1}{8} + \frac{1}{16}}{{(8 x + 1)}^{2}}$

Step 6.1.6.3

Rewrite the expression.

$\frac{\frac{1}{16} - \frac{1}{8} + \frac{1}{16}}{{(8 x + 1)}^{2}}$

Step 6.1.7

To write $- \frac{1}{8}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\frac{1}{16} - \frac{1}{8} \cdot \frac{2}{2} + \frac{1}{16}}{{(8 x + 1)}^{2}}$

Step 6.1.8

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.1.8.1

Multiply $\frac{1}{8}$ by $\frac{2}{2}$ .

$\frac{\frac{1}{16} - \frac{2}{8 \cdot 2} + \frac{1}{16}}{{(8 x + 1)}^{2}}$

Step 6.1.8.2

Multiply $8$ by $2$ .

$\frac{\frac{1}{16} - \frac{2}{16} + \frac{1}{16}}{{(8 x + 1)}^{2}}$

Step 6.1.9

Combine the numerators over the common denominator.

$\frac{\frac{1 - 2}{16} + \frac{1}{16}}{{(8 x + 1)}^{2}}$

Step 6.1.10

Subtract $2$ from $1$ .

$\frac{\frac{- 1}{16} + \frac{1}{16}}{{(8 x + 1)}^{2}}$

Step 6.1.11

Combine the numerators over the common denominator.

$\frac{\frac{- 1 + 1}{16}}{{(8 x + 1)}^{2}}$

Step 6.1.12

Add $- 1$ and $1$ .

$\frac{\frac{0}{16}}{{(8 x + 1)}^{2}}$

Step 6.1.13

Divide $0$ by $16$ .

$\frac{0}{{(8 x + 1)}^{2}}$

Step 6.2

Divide $0$ by ${(8 x + 1)}^{2}$ .

$0$