Calculus Examples

$lim_{x \to \frac{1}{2}} \frac{x^{3} + 5 x}{4 x - 6}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $\frac{1}{2}$ .

$\frac{lim_{x \to \frac{1}{2}} x^{3} + 5 x}{lim_{x \to \frac{1}{2}} 4 x - 6}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{1}{2}$ .

$\frac{lim_{x \to \frac{1}{2}} x^{3} + lim_{x \to \frac{1}{2}} 5 x}{lim_{x \to \frac{1}{2}} 4 x - 6}$

Step 3

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to \frac{1}{2}} x)}^{3} + lim_{x \to \frac{1}{2}} 5 x}{lim_{x \to \frac{1}{2}} 4 x - 6}$

Step 4

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{{(lim_{x \to \frac{1}{2}} x)}^{3} + 5 lim_{x \to \frac{1}{2}} x}{lim_{x \to \frac{1}{2}} 4 x - 6}$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{1}{2}$ .

$\frac{{(lim_{x \to \frac{1}{2}} x)}^{3} + 5 lim_{x \to \frac{1}{2}} x}{lim_{x \to \frac{1}{2}} 4 x - lim_{x \to \frac{1}{2}} 6}$

Step 6

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

$\frac{{(lim_{x \to \frac{1}{2}} x)}^{3} + 5 lim_{x \to \frac{1}{2}} x}{4 lim_{x \to \frac{1}{2}} x - lim_{x \to \frac{1}{2}} 6}$

Step 7

Evaluate the limit of $6$ which is constant as $x$ approaches $\frac{1}{2}$ .

$\frac{{(lim_{x \to \frac{1}{2}} x)}^{3} + 5 lim_{x \to \frac{1}{2}} x}{4 lim_{x \to \frac{1}{2}} x - 1 \cdot 6}$

Step 8

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

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Step 8.1

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

$\frac{{(\frac{1}{2})}^{3} + 5 lim_{x \to \frac{1}{2}} x}{4 lim_{x \to \frac{1}{2}} x - 1 \cdot 6}$

Step 8.2

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

$\frac{{(\frac{1}{2})}^{3} + 5 (\frac{1}{2})}{4 lim_{x \to \frac{1}{2}} x - 1 \cdot 6}$

Step 8.3

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

$\frac{{(\frac{1}{2})}^{3} + 5 (\frac{1}{2})}{4 (\frac{1}{2}) - 1 \cdot 6}$

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

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

$\frac{\frac{1^{3}}{2^{3}} + 5 (\frac{1}{2})}{4 (\frac{1}{2}) - 1 \cdot 6}$

Step 9.1.2

One to any power is one.

$\frac{\frac{1}{2^{3}} + 5 (\frac{1}{2})}{4 (\frac{1}{2}) - 1 \cdot 6}$

Step 9.1.3

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

$\frac{\frac{1}{8} + 5 (\frac{1}{2})}{4 (\frac{1}{2}) - 1 \cdot 6}$

Step 9.1.4

Combine $5$ and $\frac{1}{2}$ .

$\frac{\frac{1}{8} + \frac{5}{2}}{4 (\frac{1}{2}) - 1 \cdot 6}$

Step 9.1.5

To write $\frac{5}{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{\frac{1}{8} + \frac{5}{2} \cdot \frac{4}{4}}{4 (\frac{1}{2}) - 1 \cdot 6}$

Step 9.1.6

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 9.1.6.1

Multiply $\frac{5}{2}$ by $\frac{4}{4}$ .

$\frac{\frac{1}{8} + \frac{5 \cdot 4}{2 \cdot 4}}{4 (\frac{1}{2}) - 1 \cdot 6}$

Step 9.1.6.2

Multiply $2$ by $4$ .

$\frac{\frac{1}{8} + \frac{5 \cdot 4}{8}}{4 (\frac{1}{2}) - 1 \cdot 6}$

Step 9.1.7

Combine the numerators over the common denominator.

$\frac{\frac{1 + 5 \cdot 4}{8}}{4 (\frac{1}{2}) - 1 \cdot 6}$

Step 9.1.8

Simplify the numerator.

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Step 9.1.8.1

Multiply $5$ by $4$ .

$\frac{\frac{1 + 20}{8}}{4 (\frac{1}{2}) - 1 \cdot 6}$

Step 9.1.8.2

Add $1$ and $20$ .

$\frac{\frac{21}{8}}{4 (\frac{1}{2}) - 1 \cdot 6}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

Cancel the common factor of $2$ .

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Step 9.2.1.1

Factor $2$ out of $4$ .

$\frac{\frac{21}{8}}{2 (2) \frac{1}{2} - 1 \cdot 6}$

Step 9.2.1.2

Cancel the common factor.

$\frac{\frac{21}{8}}{2 \cdot 2 \frac{1}{2} - 1 \cdot 6}$

Step 9.2.1.3

Rewrite the expression.

$\frac{\frac{21}{8}}{2 - 1 \cdot 6}$

Step 9.2.2

Multiply $- 1$ by $6$ .

$\frac{\frac{21}{8}}{2 - 6}$

Step 9.2.3

Subtract $6$ from $2$ .

$\frac{\frac{21}{8}}{- 4}$

Step 9.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{21}{8} \cdot \frac{1}{- 4}$

Step 9.4

Move the negative in front of the fraction.

$\frac{21}{8} (- \frac{1}{4})$

Step 9.5

Multiply $\frac{21}{8} (- \frac{1}{4})$ .

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Step 9.5.1

Multiply $\frac{21}{8}$ by $\frac{1}{4}$ .

$- \frac{21}{8 \cdot 4}$

Step 9.5.2

Multiply $8$ by $4$ .

$- \frac{21}{32}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- \frac{21}{32}$

Decimal Form:

$- 0.65625$