Calculus Examples

$\frac{lim_{x \to 0} \tan (0.001)}{\tan (7 (0.001))}$

Step 1

Evaluate the limit of $\tan (0.001)$ which is constant as $x$ approaches $0$ .

$\frac{\tan (0.001)}{\tan (7 (0.001))}$

Step 2

Simplify the answer.

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Step 2.1

Rewrite $\tan (7 (0.001))$ in terms of sines and cosines.

$\frac{\tan (0.001)}{\frac{\sin (7 \cdot 0.001)}{\cos (7 \cdot 0.001)}}$

Step 2.2

Rewrite $\tan (0.001)$ in terms of sines and cosines.

$\frac{\frac{\sin (0.001)}{\cos (0.001)}}{\frac{\sin (7 \cdot 0.001)}{\cos (7 \cdot 0.001)}}$

Step 2.3

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (7 \cdot 0.001)}{\cos (7 \cdot 0.001)}$ .

$\frac{\sin (0.001)}{\cos (0.001)} \cdot \frac{\cos (7 \cdot 0.001)}{\sin (7 \cdot 0.001)}$

Step 2.4

Simplify.

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Step 2.4.1

Convert from $\frac{\sin (0.001)}{\cos (0.001)}$ to $\tan (0.001)$ .

$\tan (0.001) \frac{\cos (7 \cdot 0.001)}{\sin (7 \cdot 0.001)}$

Step 2.4.2

Convert from $\frac{\cos (7 \cdot 0.001)}{\sin (7 \cdot 0.001)}$ to $\cot (7 \cdot 0.001)$ .

$\tan (0.001) \cot (7 \cdot 0.001)$

Step 2.5

Multiply $7$ by $0.001$ .

$\tan (0.001) \cot (0.007)$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\tan (0.001) \cot (0.007)$

Decimal Form:

$0.14285714 \dots$