Question

Let $f(x) = \begin{cases} -2 \sin x, &x \leq - \pi /2 \\ a \sin x +b, & - \pi /2 < x < \pi /2 \\ \cos \, x , & x \geq \pi /2 \end{cases}$ then the? values of a and b so that $f(x)$ is continuous are

• A. $ a = 1, b = 1 $
• B. $a= 1,b= -1 $
• C. $a=-1 \: b=l$
• D. $a = - 1,b = - 1 $

Answer 1

The Correct Option is C

$f(x) =
\begin{cases}
-2 \sin x, &x \leq - \pi /2 \\
a \sin x +b, & - \pi /2 < x < \pi /2 \\
\cos \, x , & x \geq \pi /2
\end{cases}\(Given that\)f(x)$ is continuous. 
\(\therefore \:\: \lim _{x\to \frac{\pi ^{-}}{2}} f\left(x\right) = \lim _{x\to \frac{\pi ^{+}}{2}} f\left(x\right) = f\left(\frac{\pi}{2}\right)\)
\(\lim _{x\to \frac{\pi ^{-}}{2}} a \sin x + b = \lim _{x\to \frac{\pi ^{+}}{2}} \cos x = \cos x = \cos\left(\frac{\pi}{2}\right)\)
\(\lim_{h \rightarrow0} a \sin \left(\frac{\pi }{2}-h\right)+b = \lim _{h \rightarrow 0} \cos\left(\frac{\pi}{2}+h\right) = 0\)
\(\Rightarrow a+b =0\) ......(i) 
Now for \(x = -\frac{\pi}{2}\)
\(\lim_{x \rightarrow \frac{\pi^{-}}{2}} f\left(x\right) = \lim _{x \rightarrow \frac{\pi ^{+}}{2}} f\left(x\right) = f\left(-\frac{\pi }{2}\right)\)
\(\Rightarrow \lim _{x \rightarrow \frac{\pi ^{-}}{2}} \left(-2 \sin x \right)\)
\(= \lim _{x\to \frac{\pi ^{+}}{2}} \left(a \sin x + b\right) = - 2\sin \left(- \frac{\pi }{2}\right)\)
\(\Rightarrow \lim _{h \rightarrow 0} - 2 \sin \left(- \frac{\pi }{2} -h \right)\)
\(= \lim _{h \rightarrow 0} a \sin \left(- \frac{\pi }{2} +h \right)+ b=2\)
\(\Rightarrow 2 = -a+b =2 \Rightarrow b-a = 2\) ....(ii) 
Adding (i) and (ii) we get 
\(2b = 2\Rightarrow b = 1 \Rightarrow a =-1\) 
Hence,\(f(x)\) is continuous for \(a = -1 , b = 1\)

The ability to trace a function's graph with a pencil without taking the pencil off the paper is a feature of many functions. These are referred to as continuous functions. If a function's graph does not break at a particular point, it is said to be continuous at that location. In general, an introductory calculus course will give a precise explanation of how the limit concept applies to the continuity of a real function. First, a function f with variable x is continuous at point "a" on the real line if and only if the limit of f(x) equals the value of f(x) at "a," i.e., f(a), as x approaches "a."

Following are some mathematical definitions of continuity:

If the following three conditions are met, a function is said to be continuous at a given point.

• f(a) is defined
• lim _x→a f(x) exists
• lim_x→a f(x)=lim _x→a f(x)=f(a)

When a graph can be traced without lifting the pen from the sheet, the function is said to be a continuous function. A function, on the other hand, is said to be discontinuous if it contains any gaps in between.