Calculus Examples

$lim_{x \to 0} \frac{\cot (3 x)}{\csc (3 x)}$

Step 1

Apply trigonometric identities.

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Step 1.1

Rewrite $\csc (3 x)$ in terms of sines and cosines.

$lim_{x \to 0} \frac{\cot (3 x)}{\frac{1}{\sin (3 x)}}$

Step 1.2

Rewrite $\cot (3 x)$ in terms of sines and cosines.

$lim_{x \to 0} \frac{\frac{\cos (3 x)}{\sin (3 x)}}{\frac{1}{\sin (3 x)}}$

Step 1.3

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (3 x)}$ .

$lim_{x \to 0} \frac{\cos (3 x)}{\sin (3 x)} \sin (3 x)$

Step 1.4

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

$lim_{x \to 0} \frac{\cos (3 x)}{\sin (3 x)} \cdot \frac{\sin (3 x)}{1}$

Step 1.5

Cancel the common factor of $\sin (3 x)$ .

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Step 1.5.1

Cancel the common factor.

$lim_{x \to 0} \frac{\cos (3 x)}{\sin (3 x)} \cdot \frac{\sin (3 x)}{1}$

Step 1.5.2

Rewrite the expression.

$lim_{x \to 0} \cos (3 x)$

Step 2

Evaluate the limit.

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Step 2.1

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

$\cos (lim_{x \to 0} 3 x)$

Step 2.2

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

$\cos (3 lim_{x \to 0} x)$

Step 3

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\cos (3 \cdot 0)$

Step 4

Simplify the answer.

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Step 4.1

Multiply $3$ by $0$ .

$\cos (0)$

Step 4.2

The exact value of $\cos (0)$ is $1$ .

$1$