Calculus Examples

$lim_{x \to 8} \frac{\sin (\frac{3}{x})}{\tan (\frac{6}{x})}$

Step 1

Apply trigonometric identities.

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Step 1.1

Rewrite $\tan (\frac{6}{x})$ in terms of sines and cosines.

$lim_{x \to 8} \frac{\sin (\frac{3}{x})}{\frac{\sin (\frac{6}{x})}{\cos (\frac{6}{x})}}$

Step 1.2

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (\frac{6}{x})}{\cos (\frac{6}{x})}$ .

$lim_{x \to 8} \frac{\sin (\frac{3}{x}) \cos (\frac{6}{x})}{\sin (\frac{6}{x})}$

Step 1.3

Convert from $\frac{\sin (\frac{3}{x}) \cos (\frac{6}{x})}{\sin (\frac{6}{x})}$ to $\sin (\frac{3}{x}) \cot (\frac{6}{x})$ .

$lim_{x \to 8} \sin (\frac{3}{x}) \cot (\frac{6}{x})$

Step 2

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

$lim_{x \to 8} \sin (\frac{3}{x}) \cdot lim_{x \to 8} \cot (\frac{6}{x})$

Step 3

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

$\sin (lim_{x \to 8} \frac{3}{x}) \cdot lim_{x \to 8} \cot (\frac{6}{x})$

Step 4

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

$\sin (3 lim_{x \to 8} \frac{1}{x}) \cdot lim_{x \to 8} \cot (\frac{6}{x})$

Step 5

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

$\sin (3 \frac{lim_{x \to 8} 1}{lim_{x \to 8} x}) \cdot lim_{x \to 8} \cot (\frac{6}{x})$

Step 6

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

$\sin (3 \frac{1}{lim_{x \to 8} x}) \cdot lim_{x \to 8} \cot (\frac{6}{x})$

Step 7

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

$\sin (3 \frac{1}{lim_{x \to 8} x}) \cdot \cot (lim_{x \to 8} \frac{6}{x})$

Step 8

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\sin (3 \frac{1}{lim_{x \to 8} x}) \cdot \cot (6 lim_{x \to 8} \frac{1}{x})$

Step 9

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

$\sin (3 \frac{1}{lim_{x \to 8} x}) \cdot \cot (6 \frac{lim_{x \to 8} 1}{lim_{x \to 8} x})$

Step 10

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

$\sin (3 \frac{1}{lim_{x \to 8} x}) \cdot \cot (6 \frac{1}{lim_{x \to 8} x})$

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 $x$ .

$\sin (3 (\frac{1}{8})) \cdot \cot (6 \frac{1}{lim_{x \to 8} x})$

Step 11.2

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

$\sin (3 (\frac{1}{8})) \cdot \cot (6 (\frac{1}{8}))$

Step 12

Simplify the answer.

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Step 12.1

Combine $3$ and $\frac{1}{8}$ .

$\sin (\frac{3}{8}) \cdot \cot (6 (\frac{1}{8}))$

Step 12.2

Evaluate $\sin (\frac{3}{8})$ .

$0.36627252 \cdot \cot (6 (\frac{1}{8}))$

Step 12.3

Cancel the common factor of $2$ .

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Step 12.3.1

Factor $2$ out of $6$ .

$0.36627252 \cdot \cot (2 (3) \frac{1}{8})$

Step 12.3.2

Factor $2$ out of $8$ .

$0.36627252 \cdot \cot (2 \cdot 3 \frac{1}{2 \cdot 4})$

Step 12.3.3

Cancel the common factor.

$0.36627252 \cdot \cot (2 \cdot 3 \frac{1}{2 \cdot 4})$

Step 12.3.4

Rewrite the expression.

$0.36627252 \cdot \cot (3 (\frac{1}{4}))$

Step 12.4

Combine $3$ and $\frac{1}{4}$ .

$0.36627252 \cdot \cot (\frac{3}{4})$

Step 12.5

Evaluate $\cot (\frac{3}{4})$ .

$0.36627252 \cdot 1.07342614$

Step 12.6

Multiply $0.36627252$ by $1.07342614$ .

$0.39316651$