Calculus Examples

$lim_{x \to 8} \frac{1 - (\cos^{2} (3 x))}{x \sin (5 x)}$

Step 1

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

$\frac{lim_{x \to 8} 1 - (\cos^{2} (3 x))}{lim_{x \to 8} x \sin (5 x)}$

Step 2

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

$\frac{lim_{x \to 8} 1 - lim_{x \to 8} \cos^{2} (3 x)}{lim_{x \to 8} x \sin (5 x)}$

Step 3

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

$\frac{1 - lim_{x \to 8} \cos^{2} (3 x)}{lim_{x \to 8} x \sin (5 x)}$

Step 4

Move the exponent $2$ from $\cos^{2} (3 x)$ outside the limit using the Limits Power Rule.

$\frac{1 - {(lim_{x \to 8} \cos (3 x))}^{2}}{lim_{x \to 8} x \sin (5 x)}$

Step 5

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

$\frac{1 - \cos^{2} (lim_{x \to 8} 3 x)}{lim_{x \to 8} x \sin (5 x)}$

Step 6

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

$\frac{1 - \cos^{2} (3 lim_{x \to 8} x)}{lim_{x \to 8} x \sin (5 x)}$

Step 7

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

$\frac{1 - \cos^{2} (3 lim_{x \to 8} x)}{lim_{x \to 8} x \cdot lim_{x \to 8} \sin (5 x)}$

Step 8

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

$\frac{1 - \cos^{2} (3 lim_{x \to 8} x)}{lim_{x \to 8} x \cdot \sin (lim_{x \to 8} 5 x)}$

Step 9

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

$\frac{1 - \cos^{2} (3 lim_{x \to 8} x)}{lim_{x \to 8} x \cdot \sin (5 lim_{x \to 8} x)}$

Step 10

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

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Step 10.1

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

$\frac{1 - \cos^{2} (3 \cdot 8)}{lim_{x \to 8} x \cdot \sin (5 lim_{x \to 8} x)}$

Step 10.2

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

$\frac{1 - \cos^{2} (3 \cdot 8)}{8 \cdot \sin (5 lim_{x \to 8} x)}$

Step 10.3

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

$\frac{1 - \cos^{2} (3 \cdot 8)}{8 \cdot \sin (5 \cdot 8)}$

Step 11

Simplify the answer.

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Step 11.1

Apply pythagorean identity.

$\frac{\sin^{2} (3 \cdot 8)}{8 \cdot \sin (5 \cdot 8)}$

Step 11.2

Factor $\sin (3 \cdot 8)$ out of $\sin^{2} (3 \cdot 8)$ .

$\frac{\sin (3 \cdot 8) \sin (3 \cdot 8)}{8 \cdot \sin (5 \cdot 8)}$

Step 11.3

Separate fractions.

$\frac{\sin (3 \cdot 8)}{8} \cdot \frac{\sin (3 \cdot 8)}{\sin (5 \cdot 8)}$

Step 11.4

Rewrite $\frac{\sin (3 \cdot 8)}{\sin (5 \cdot 8)}$ as a product.

$\frac{\sin (3 \cdot 8)}{8} (\sin (3 \cdot 8) \frac{1}{\sin (5 \cdot 8)})$

Step 11.5

Write $\sin (3 \cdot 8)$ as a fraction with denominator $1$ .

$\frac{\sin (3 \cdot 8)}{8} (\frac{\sin (3 \cdot 8)}{1} \cdot \frac{1}{\sin (5 \cdot 8)})$

Step 11.6

Simplify.

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Step 11.6.1

Divide $\sin (3 \cdot 8)$ by $1$ .

$\frac{\sin (3 \cdot 8)}{8} (\sin (3 \cdot 8) \frac{1}{\sin (5 \cdot 8)})$

Step 11.6.2

Convert from $\frac{1}{\sin (5 \cdot 8)}$ to $\csc (5 \cdot 8)$ .

$\frac{\sin (3 \cdot 8)}{8} (\sin (3 \cdot 8) \csc (5 \cdot 8))$

Step 11.7

Simplify the numerator.

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Step 11.7.1

Multiply $3$ by $8$ .

$\frac{\sin (24)}{8} (\sin (3 \cdot 8) \csc (5 \cdot 8))$

Step 11.7.2

Evaluate $\sin (24)$ .

$\frac{0.40673664}{8} (\sin (3 \cdot 8) \csc (5 \cdot 8))$

Step 11.8

Divide $0.40673664$ by $8$ .

$0.05084208 (\sin (3 \cdot 8) \csc (5 \cdot 8))$

Step 11.9

Multiply $3$ by $8$ .

$0.05084208 (\sin (24) \csc (5 \cdot 8))$

Step 11.10

Evaluate $\sin (24)$ .

$0.05084208 (0.40673664 \csc (5 \cdot 8))$

Step 11.11

Multiply $5$ by $8$ .

$0.05084208 (0.40673664 \csc (40))$

Step 11.12

Evaluate $\csc (40)$ .

$0.05084208 (0.40673664 \cdot 1.55572382)$

Step 11.13

Multiply $0.05084208 (0.40673664 \cdot 1.55572382)$ .

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Step 11.13.1

Multiply $0.40673664$ by $1.55572382$ .

$0.05084208 \cdot 0.63276988$

Step 11.13.2

Multiply $0.05084208$ by $0.63276988$ .

$0.03217133$