Calculus Examples

$lim_{x \to 8} 2 + \frac{132}{x} + \frac{\sin^{4} (x^{5})}{x^{4}}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$lim_{x \to 8} 2 + lim_{x \to 8} \frac{132}{x} + lim_{x \to 8} \frac{\sin^{4} (x^{5})}{x^{4}}$

Step 2

Evaluate the limit of $2$ which is constant as $x$ approaches $8$ .

$2 + lim_{x \to 8} \frac{132}{x} + lim_{x \to 8} \frac{\sin^{4} (x^{5})}{x^{4}}$

Step 3

Move the term $132$ outside of the limit because it is constant with respect to $x$ .

$2 + 132 lim_{x \to 8} \frac{1}{x} + lim_{x \to 8} \frac{\sin^{4} (x^{5})}{x^{4}}$

Step 4

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $8$ .

$2 + 132 \frac{lim_{x \to 8} 1}{lim_{x \to 8} x} + lim_{x \to 8} \frac{\sin^{4} (x^{5})}{x^{4}}$

Step 5

Evaluate the limit of $1$ which is constant as $x$ approaches $8$ .

$2 + 132 \frac{1}{lim_{x \to 8} x} + lim_{x \to 8} \frac{\sin^{4} (x^{5})}{x^{4}}$

Step 6

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $8$ .

$2 + 132 \frac{1}{lim_{x \to 8} x} + \frac{lim_{x \to 8} \sin^{4} (x^{5})}{lim_{x \to 8} x^{4}}$

Step 7

Move the exponent $4$ from $\sin^{4} (x^{5})$ outside the limit using the Limits Power Rule.

$2 + 132 \frac{1}{lim_{x \to 8} x} + \frac{{(lim_{x \to 8} \sin (x^{5}))}^{4}}{lim_{x \to 8} x^{4}}$

Step 8

Move the limit inside the trig function because sine is continuous.

$2 + 132 \frac{1}{lim_{x \to 8} x} + \frac{\sin^{4} (lim_{x \to 8} x^{5})}{lim_{x \to 8} x^{4}}$

Step 9

Move the exponent $5$ from $x^{5}$ outside the limit using the Limits Power Rule.

$2 + 132 \frac{1}{lim_{x \to 8} x} + \frac{\sin^{4} ({(lim_{x \to 8} x)}^{5})}{lim_{x \to 8} x^{4}}$

Step 10

Move the exponent $4$ from $x^{4}$ outside the limit using the Limits Power Rule.

$2 + 132 \frac{1}{lim_{x \to 8} x} + \frac{\sin^{4} ({(lim_{x \to 8} x)}^{5})}{{(lim_{x \to 8} x)}^{4}}$

Step 11

Evaluate the limits by plugging in $8$ for all occurrences of $x$ .

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Step 11.1

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$2 + 132 (\frac{1}{8}) + \frac{\sin^{4} ({(lim_{x \to 8} x)}^{5})}{{(lim_{x \to 8} x)}^{4}}$

Step 11.2

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$2 + 132 (\frac{1}{8}) + \frac{\sin^{4} (8^{5})}{{(lim_{x \to 8} x)}^{4}}$

Step 11.3

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$2 + 132 (\frac{1}{8}) + \frac{\sin^{4} (8^{5})}{8^{4}}$

Step 12

Simplify the answer.

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Step 12.1

Simplify each term.

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Step 12.1.1

Cancel the common factor of $4$ .

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Step 12.1.1.1

Factor $4$ out of $132$ .

$2 + 4 (33) \frac{1}{8} + \frac{\sin^{4} (8^{5})}{8^{4}}$

Step 12.1.1.2

Factor $4$ out of $8$ .

$2 + 4 \cdot 33 \frac{1}{4 \cdot 2} + \frac{\sin^{4} (8^{5})}{8^{4}}$

Step 12.1.1.3

Cancel the common factor.

$2 + 4 \cdot 33 \frac{1}{4 \cdot 2} + \frac{\sin^{4} (8^{5})}{8^{4}}$

Step 12.1.1.4

Rewrite the expression.

$2 + 33 (\frac{1}{2}) + \frac{\sin^{4} (8^{5})}{8^{4}}$

Step 12.1.2

Combine $33$ and $\frac{1}{2}$ .

$2 + \frac{33}{2} + \frac{\sin^{4} (8^{5})}{8^{4}}$

Step 12.1.3

Simplify the numerator.

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Step 12.1.3.1

Raise $8$ to the power of $5$ .

$2 + \frac{33}{2} + \frac{\sin^{4} (32768)}{8^{4}}$

Step 12.1.3.2

Remove full rotations of $360$ ° until the angle is between $0$ ° and $360$ °.

$2 + \frac{33}{2} + \frac{\sin^{4} (8)}{8^{4}}$

Step 12.1.3.3

Evaluate $\sin (8)$ .

$2 + \frac{33}{2} + \frac{{0.1391731}^{4}}{8^{4}}$

Step 12.1.3.4

Raise $0.1391731$ to the power of $4$ .

$2 + \frac{33}{2} + \frac{0.00037516}{8^{4}}$

Step 12.1.4

Raise $8$ to the power of $4$ .

$2 + \frac{33}{2} + \frac{0.00037516}{4096}$

Step 12.1.5

Divide $0.00037516$ by $4096$ .

$2 + \frac{33}{2} + 0.00000009$

Step 12.2

Find the common denominator.

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Step 12.2.1

Write $2$ as a fraction with denominator $1$ .

$\frac{2}{1} + \frac{33}{2} + 0.00000009$

Step 12.2.2

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

$\frac{2}{1} \cdot \frac{2}{2} + \frac{33}{2} + 0.00000009$

Step 12.2.3

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

$\frac{2 \cdot 2}{2} + \frac{33}{2} + 0.00000009$

Step 12.2.4

Write $0.00000009$ as a fraction with denominator $1$ .

$\frac{2 \cdot 2}{2} + \frac{33}{2} + \frac{0.00000009}{1}$

Step 12.2.5

Multiply $\frac{0.00000009}{1}$ by $\frac{2}{2}$ .

$\frac{2 \cdot 2}{2} + \frac{33}{2} + \frac{0.00000009}{1} \cdot \frac{2}{2}$

Step 12.2.6

Multiply $\frac{0.00000009}{1}$ by $\frac{2}{2}$ .

$\frac{2 \cdot 2}{2} + \frac{33}{2} + \frac{0.00000009 \cdot 2}{2}$

Step 12.3

Combine the numerators over the common denominator.

$\frac{2 \cdot 2 + 33 + 0.00000009 \cdot 2}{2}$

Step 12.4

Simplify each term.

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Step 12.4.1

Multiply $2$ by $2$ .

$\frac{4 + 33 + 0.00000009 \cdot 2}{2}$

Step 12.4.2

Multiply $0.00000009$ by $2$ .

$\frac{4 + 33 + 0.00000018}{2}$

Step 12.5

Add $4$ and $33$ .

$\frac{37 + 0.00000018}{2}$

Step 12.6

Add $37$ and $0.00000018$ .

$\frac{37.00000018}{2}$

Step 12.7

Divide $37.00000018$ by $2$ .

$18.50000009$