Calculus Examples

$\frac{lim_{x \to \frac{π}{4}} \sin (x)}{\tan (x) - 1}$

Step 1

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

$\frac{\sin (lim_{x \to \frac{π}{4}} x)}{\tan (x) - 1}$

Step 2

Evaluate the limit of $x$ by plugging in $\frac{π}{4}$ for $x$ .

$\frac{\sin (\frac{π}{4})}{\tan (x) - 1}$

Step 3

Simplify the answer.

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Step 3.1

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\frac{\sqrt{2}}{2}}{\tan (x) - 1}$

Step 3.2

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

$\frac{\frac{\sqrt{2}}{2}}{\frac{\sin (x)}{\cos (x)} - 1}$

Step 3.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{\sqrt{2}}{2} \cdot \frac{1}{\frac{\sin (x)}{\cos (x)} - 1}$

Step 3.4

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{1}{\frac{\sin (x)}{\cos (x)} - 1}$ .

$\frac{\sqrt{2}}{2 (\frac{\sin (x)}{\cos (x)} - 1)}$

Step 3.5

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

$\frac{\sqrt{2}}{2 (\tan (x) - 1)}$