What is [tex]\lim _{x \rightarrow-1}\left(\sin \left(\frac{\pi x}{6}\right)\right)[/tex]?

A. [tex]-\frac{\sqrt{3}}{2}[/tex]
B. [tex]-\frac{1}{2}[/tex]
C. [tex]\frac{1}{2}[/tex]
D. [tex]\frac{\sqrt{3}}{2}[/tex]



Answer :

Let's find the limit of the function [tex]\(\sin \left( \frac{\pi x}{6} \right)\)[/tex] as [tex]\(x\)[/tex] approaches -1.

To determine this limit, we can directly substitute [tex]\(x = -1\)[/tex] into the function and evaluate. Since the sine function is continuous, we can use substitution for the limit:

[tex]\[ \lim_{x \to -1} \sin \left( \frac{\pi x}{6} \right) = \sin \left( \frac{\pi (-1)}{6} \right) \][/tex]

Now, we compute the argument of the sine function:

[tex]\[ \frac{\pi (-1)}{6} = -\frac{\pi}{6} \][/tex]

Next, we need to find the sine of [tex]\(-\frac{\pi}{6}\)[/tex]. Recall that the sine function has the property:

[tex]\[ \sin(-\theta) = -\sin(\theta) \][/tex]

So,

[tex]\[ \sin\left(-\frac{\pi}{6}\right) = -\sin\left(\frac{\pi}{6}\right) \][/tex]

We know from trigonometry that:

[tex]\[ \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \][/tex]

Thus,

[tex]\[ \sin\left(-\frac{\pi}{6}\right) = -\frac{1}{2} \][/tex]

Therefore, the limit is:

[tex]\[ \lim_{x \to -1} \sin \left( \frac{\pi x}{6} \right) = -\frac{1}{2} \][/tex]

Given the options:

[tex]\[ -\frac{\sqrt{3}}{2}, \quad -\frac{1}{2}, \quad \frac{1}{2}, \quad \frac{\sqrt{3}}{2} \][/tex]

The correct answer is [tex]\(\boxed{-\frac{1}{2}}\)[/tex].