Question

If $f\left(x\right)=\{\begin{array}{c}\begin{array}{cc}x+\lambda ,& -1<x<3\end{array}\\ \begin{array}{cc}4,& x=3\end{array}\\ \begin{array}{cc}3x-5,& x>3\end{array}\end{array}$ is continuous at $x=3$ then tha value of $\lambda$ is

• A. $-1$
• B. $0$
• C. $1$
• D. does not exits

Answer 1

Answer: (C)

$1$

Function $f\left(x\right)=\{\begin{array}{c}\begin{array}{cc}x+\lambda ,& -1>x>3\end{array}\\ \begin{array}{cc}4,& x=3\end{array}\\ \begin{array}{cc}3x-5,& x>3\end{array}\end{array}$ is continuous at $x=3$.

We know that as $f\left(x\right)$ is continuous at $x=a,$ therefore $\underset{x\to a}{lim}f\left(x\right)=f\left(a\right)$.

Since the given function is continuous at $x=3,$

therefore $\underset{x\to 3-}{lim}(x+\lambda )=f\left(3\right)\Rightarrow 3+\lambda =4\Rightarrow \lambda =4-3=1$