Calculus Examples

$lim_{x \to 0} \frac{\tan (- 0.1)}{\tan (5 (- 0.1))}$

Step 1

Apply trigonometric identities.

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Step 1.1

Rewrite $\tan (5 (- 0.1))$ in terms of sines and cosines.

$lim_{x \to 0} \frac{\tan (- 0.1)}{\frac{\sin (5 \cdot - 0.1)}{\cos (5 \cdot - 0.1)}}$

Step 1.2

Rewrite $\tan (- 0.1)$ in terms of sines and cosines.

$lim_{x \to 0} \frac{\frac{\sin (- 0.1)}{\cos (- 0.1)}}{\frac{\sin (5 \cdot - 0.1)}{\cos (5 \cdot - 0.1)}}$

Step 1.3

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (5 \cdot - 0.1)}{\cos (5 \cdot - 0.1)}$ .

$lim_{x \to 0} \frac{\sin (- 0.1)}{\cos (- 0.1)} \cdot \frac{\cos (5 \cdot - 0.1)}{\sin (5 \cdot - 0.1)}$

Step 1.4

Simplify.

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Step 1.4.1

Convert from $\frac{\sin (- 0.1)}{\cos (- 0.1)}$ to $\tan (- 0.1)$ .

$lim_{x \to 0} \tan (- 0.1) \frac{\cos (5 \cdot - 0.1)}{\sin (5 \cdot - 0.1)}$

Step 1.4.2

Convert from $\frac{\cos (5 \cdot - 0.1)}{\sin (5 \cdot - 0.1)}$ to $\cot (5 \cdot - 0.1)$ .

$lim_{x \to 0} \tan (- 0.1) \cot (5 \cdot - 0.1)$

Step 2

Evaluate the limit of $\tan (- 0.1) \cot (5 \cdot - 0.1)$ which is constant as $x$ approaches $0$ .

$\tan (- 0.1) \cot (5 \cdot - 0.1)$

Step 3

Multiply $5$ by $- 0.1$ .

$\tan (- 0.1) \cot (- 0.5)$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\tan (- 0.1) \cot (- 0.5)$

Decimal Form:

$0.18366138 \dots$