Calculus Examples

$lim_{x \to 0} \frac{\tan (10 t)}{\sin (2 t)}$

Step 1

Apply trigonometric identities.

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Step 1.1

Rewrite $\tan (10 t)$ in terms of sines and cosines.

$lim_{x \to 0} \frac{\frac{\sin (10 t)}{\cos (10 t)}}{\sin (2 t)}$

Step 1.2

Rewrite $\frac{\frac{\sin (10 t)}{\cos (10 t)}}{\sin (2 t)}$ as a product.

$lim_{x \to 0} \frac{\sin (10 t)}{\cos (10 t)} \cdot \frac{1}{\sin (2 t)}$

Step 1.3

Simplify.

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Step 1.3.1

Convert from $\frac{\sin (10 t)}{\cos (10 t)}$ to $\tan (10 t)$ .

$lim_{x \to 0} \tan (10 t) \frac{1}{\sin (2 t)}$

Step 1.3.2

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

$lim_{x \to 0} \tan (10 t) \csc (2 t)$

Step 2

Evaluate the limit of $\tan (10 t) \csc (2 t)$ which is constant as $x$ approaches $0$ .

$\tan (10 t) \csc (2 t)$