Calculus Examples

$\frac{lim_{x \to \frac{π}{6}} 1 - \cot (x)}{\sin (x) - \cos (x)}$

Step 1

Evaluate the limit.

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Step 1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{π}{6}$ .

$\frac{lim_{x \to \frac{π}{6}} 1 - lim_{x \to \frac{π}{6}} \cot (x)}{\sin (x) - \cos (x)}$

Step 1.2

Evaluate the limit of $1$ which is constant as $x$ approaches $\frac{π}{6}$ .

$\frac{1 - lim_{x \to \frac{π}{6}} \cot (x)}{\sin (x) - \cos (x)}$

Step 1.3

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

$\frac{1 - \cot (lim_{x \to \frac{π}{6}} x)}{\sin (x) - \cos (x)}$

Step 2

Evaluate the limit of $x$ by plugging in $\frac{π}{6}$ for $x$ .

$\frac{1 - \cot (\frac{π}{6})}{\sin (x) - \cos (x)}$

Step 3

The exact value of $\cot (\frac{π}{6})$ is $\sqrt{3}$ .

$\frac{1 - \sqrt{3}}{\sin (x) - \cos (x)}$