Question

If $f(x)=\{\begin{array}{ll}x+\lambda & x<3\\ 4& x=3\\ 3x-5,& x>3\end{array}$
is continuous at $x=3$, then the value of $\lambda$ is equal to :

• A. 1
• B. -1
• C. 0
• D. does not exist

Answer 1

Answer: (A)

Note: A function $f(x)$ is said to be continuous at
$x=a$ iff. $R.H.L=L.H.L.=f(a)$
i.e., $\underset{x\to a^{+}}{\lim }f(x)=\underset{x\to a^{-}}{\lim }f(x)=\underset{x\to a}{\lim }f(x)$
$f(x)=\{\begin{array}{ll}x+\lambda & x<3\\ 4& x=3\\ 3x-5,& x>3\end{array}$
continuous at $x=3$
$\therefore \underset{x\to 3^{-}}{\lim }(x+\lambda )=f(3)=\underset{x\to 3^{+}}{\lim }(3x-5)$
$\Rightarrow 3+\lambda =4$
$\Rightarrow \lambda =4-3=1$