Calculus Examples

$lim_{x \to \frac{π}{6}} \frac{\sin (x) - \frac{1}{2}}{x - \frac{π}{6}}$

Step 1

Combine terms.

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Step 1.1

To write $x$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$lim_{x \to \frac{π}{6}} \frac{\sin (x) - \frac{1}{2}}{x \cdot \frac{6}{6} - \frac{π}{6}}$

Step 1.2

Combine $x$ and $\frac{6}{6}$ .

$lim_{x \to \frac{π}{6}} \frac{\sin (x) - \frac{1}{2}}{\frac{x \cdot 6}{6} - \frac{π}{6}}$

Step 1.3

Combine the numerators over the common denominator.

$lim_{x \to \frac{π}{6}} \frac{\sin (x) - \frac{1}{2}}{\frac{x \cdot 6 - π}{6}}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to \frac{π}{6}} (\sin (x) - \frac{1}{2}) \frac{6}{x \cdot 6 - π}$

Step 3

Consider the left sided limit.

$lim_{x \to {(\frac{π}{6})}^{-}} (\sin (x) - \frac{1}{2}) \frac{6}{x \cdot 6 - π}$

Step 4

Make a table to show the behavior of the function $(\sin (x) - \frac{1}{2}) \frac{6}{x \cdot 6 - π}$ as $x$ approaches $\frac{π}{6}$ from the left.

$\begin{array}{c} x & (\sin (x) - \frac{1}{2}) \frac{6}{x \cdot 6 - π} \\ 0.42359877 & 0.88956192 \\ 0.51359877 & 0.86851094 \\ 0.52259877 & 0.86627525 \\ 0.52349877 & 0.8660504 \end{array}$

Step 5

As the $x$ values approach $\frac{π}{6}$ , the function values approach $0.866$ . Thus, the limit of $(\sin (x) - \frac{1}{2}) \frac{6}{x \cdot 6 - π}$ as $x$ approaches $\frac{π}{6}$ from the left is $0.866$ .

$0.866$

Step 6

Consider the right sided limit.

$lim_{x \to {(\frac{π}{6})}^{+}} (\sin (x) - \frac{1}{2}) \frac{6}{x \cdot 6 - π}$

Step 7

Make a table to show the behavior of the function $(\sin (x) - \frac{1}{2}) \frac{6}{x \cdot 6 - π}$ as $x$ approaches $\frac{π}{6}$ from the right.

$\begin{array}{c} x & (\sin (x) - \frac{1}{2}) \frac{6}{x \cdot 6 - π} \\ 0.62359877 & 0.83960357 \\ 0.53359877 & 0.86351099 \\ 0.52459877 & 0.86577525 \\ 0.52369877 & 0.8660004 \end{array}$

Step 8

$0.866$