Question

Examine the continuity or discontinuity of the given function:

Answer 1

Given Question :- Examine the continuity or discontinuity of

$\begin{gathered}\begin{gathered}\bf\: f(x) = \begin{cases} &\sf{5x + 2, \: \: x \leqslant 0} \\ &\sf{ \: \: \: \: \: \: 6x, \: \: x > 0} \end{cases}\end{gathered}\end{gathered}$

$\large\underline{\sf{Solution-}}$

Given function is

$\begin{gathered}\begin{gathered}\bf\: f(x) = \begin{cases} &\sf{5x + 2, \: \: x \leqslant 0} \\ &\sf{ \: \: \: \: \: \: 6x, \: \: x > 0} \end{cases}\end{gathered}\end{gathered}$

Consider $\rm \: f(0) = 5 \times 0 + 2 = 2$

Consider, Left Hand Limit

$\rm \: \displaystyle\lim_{x \to 0^-}\rm f(x)$

$\rm \: = \: \displaystyle\lim_{x \to 0^-}\rm (5x + 2)$

To evaluate this limit, we use method of Substitution.

$\rm \:Put\:x=0-h,\:\: as\:\: x \to 0, \:\:so\:\:h \to 0$

$\rm \: = \: \displaystyle\lim_{h \to 0}\rm (5(- h) + 2)$

$\rm \: = \: 0 + 2$

$\rm \: = \: 2$

So,

$\rm\implies \:\displaystyle\lim_{x \to 0^-}\rm f(x) \: = \: 2$

Consider, Right Hand Limit

$\rm \: \displaystyle\lim_{x \to 0^ + }\rm f(x)$

$\rm \: = \: \displaystyle\lim_{x \to 0^ + }\rm 6x$

So, to evaluate this limit, we use method of Substitution.

$\rm \:Put\:x=0+h,\:\: as\:\:x \to 0, \:\:so\:\:h \to 0$

$\rm \: = \: \displaystyle\lim_{h \to 0 }\rm 6(h)$

$\rm \: = \: 0$

$\rm\implies \:\displaystyle\lim_{x \to 0^ + }\rm f(x) \: = \: 0$

So, from above we concluded that

$\rm\implies \:\rm \: f(0) = \displaystyle\lim_{x \to 0^-}\rm f(x) \: \ne \: \displaystyle\lim_{x \to 0^ + }\rm f(x)$

$\rm\implies \:\rm \: f(x) \: is \: not \: continuous \: at \: x \: = \: 0$