Q. 77 - Calculus

Question

Describe the intervals on which each function f  is continuous. At each point where f fails to be continuous, use limits to determine the type of discontinuity and any left- or right-continuity.

$f\left(x\right)=\left\{\begin{array}{ll}\sin x,& if x<\mathrm{\pi }\\ \cos x,& if x\ge \mathrm{\pi }\end{array}\right.$

Step-by-Step Solution

Answer

The function is continuous on $\left(-\infty ,\mathrm{\pi }\right),\left(\mathrm{\pi },\infty \right)$ and only the right limit is continuous at $x=\mathrm{\pi }$.

Step 1. Given information.

Consider the given question,

$f\left(x\right)=\left\{\begin{array}{ll}\sin x,& if x<\mathrm{\pi }\\ \cos x,& if x\ge \mathrm{\pi }\end{array}\right.$

Step 2. Calculate the left and right limits.

We need to calculate the left and right limits,

$$\begin{gathered} \underset{x\to {\mathrm{\pi }}^{+}}{\lim }f\left(x\right)=\underset{x\to {\mathrm{\pi }}^{+}}{\lim }\left(\sin x\right) \\ =\sin \left(\mathrm{\pi }\right) \\ =0 \\ \underset{x\to \mathrm{\pi }}{\lim }f\left(x\right)=\underset{x\to \mathrm{\pi }}{\lim }\left(\cos x\right) \\ =\cos \left(\mathrm{\pi }\right) \\ =-1 \\ \underset{x\to {\mathrm{\pi }}^{-}}{\lim }f\left(x\right)=\underset{x\to {\mathrm{\pi }}^{-}}{\lim }\left(\cos x\right) \\ =\cos \left(\mathrm{\pi }\right) \\ =-1 \end{gathered}$$

As the left and right  limits both exists but not equal.

Hence, the function is continuous on $\left(-\infty ,\mathrm{\pi }\right),\left(\mathrm{\pi },\infty \right)$ and only the right limit is continuous at $x=\mathrm{\pi }$.