Calculus Examples

$\frac{lim_{x \to 0} \tan (6 t)}{\sin (2 t)}$

Step 1

Evaluate the limit of $\tan (6 t)$ which is constant as $x$ approaches $0$ .

$\frac{\tan (6 t)}{\sin (2 t)}$

Step 2

Simplify the answer.

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Step 2.1

Rewrite $\tan (6 t)$ in terms of sines and cosines.

$\frac{\frac{\sin (6 t)}{\cos (6 t)}}{\sin (2 t)}$

Step 2.2

Rewrite $\frac{\frac{\sin (6 t)}{\cos (6 t)}}{\sin (2 t)}$ as a product.

$\frac{\sin (6 t)}{\cos (6 t)} \cdot \frac{1}{\sin (2 t)}$

Step 2.3

Multiply $\frac{\sin (6 t)}{\cos (6 t)}$ by $\frac{1}{\sin (2 t)}$ .

$\frac{\sin (6 t)}{\cos (6 t) \sin (2 t)}$

Step 2.4

Separate fractions.

$\frac{1}{\sin (2 t)} \cdot \frac{\sin (6 t)}{\cos (6 t)}$

Step 2.5

Convert from $\frac{\sin (6 t)}{\cos (6 t)}$ to $\tan (6 t)$ .

$\frac{1}{\sin (2 t)} \tan (6 t)$

Step 2.6

Convert from $\frac{1}{\sin (2 t)}$ to $\csc (2 t)$ .

$\csc (2 t) \tan (6 t)$