Calculus Examples

$lim_{x \to π} \frac{\sin (x)}{\tan (x)}$

Step 1

Apply trigonometric identities.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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Step 1.4.1

Cancel the common factor.

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

Step 1.4.2

Rewrite the expression.

$lim_{x \to π} \cos (x)$

Step 2

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

$\cos (lim_{x \to π} x)$

Step 3

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

$\cos (π)$

Step 4

Simplify the answer.

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Step 4.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$- \cos (0)$

Step 4.2

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

$- 1 \cdot 1$

Step 4.3

Multiply $- 1$ by $1$ .

$- 1$