Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • NCERT
  • Find which of the functions is continuous or discontinuous at the indicated points: f(x) equals 3x plus 5 if x >=2

Find which of the functions is continuous or discontinuous at the indicated points:
f(x)=\left\{\begin{array}{cl} 3 x+5, & \text { if } x \geq 2 \\ x^{2}, & \text { if } x<2 \end{array}\right. \text { at } x=2

Answers (1)

Given,

f(x)=\left\{\begin{array}{cl} 3 x+5, & \text { if } x \geq 2 \\ x^{2}, & \text { if } x<2 \end{array}\right.

We need to check its continuity at x = 2

A function f(x) is said to be continuous at x = c if,

Left hand limit (LHL at x = c) = Right hand limit (RHL at x = c) = f(c).

Mathematically we can represent it as-

\lim _{h \rightarrow 0} f(c-h)=\lim _{h \rightarrow 0} f(c+h)=f(c)

Where h is a very small number very close to 0 (h→0)

Now according to above theory-

f(x) is continuous at x = 2 if -

 

\begin{aligned} &\lim_{h \rightarrow 0} f(2-h)=\lim_{h \rightarrow 0} f(2+h)=f(2) \\ &\text{Now we can see that,} \\ &LHL=\lim_{h \rightarrow 0} f(2-h)=\lim_{h \rightarrow 0}(2-h)^{2} \text{ (using equation 1)} \\ &\therefore LHL=(2-0)^{2}=4 \ldots(2) \end{aligned}

Similarly, we proceed for RHL-

\begin{aligned} &\lim _{\mathrm{RHL}}=\underset{\mathrm{h} \rightarrow 0}{\mathrm{f}(2+\mathrm{h})}=\lim _{\mathrm{h} \rightarrow 0}\{3(2+\mathrm{h})+5\}\\ &\therefore \mathrm{RHL}=3(2+0)+5=11 \ldots(3)\\ &\text { then, }\\ &f(2)=3(2)+5=11 \ldots(4) \end{aligned}

Now from equation 2, 3 and 4 we can conclude that

\begin{aligned} &\lim _{h \rightarrow 0} f(2-h) \neq \lim _{h \rightarrow 0} f(2+h)\\ &\therefore f(x) \text { is discontinuous at } x=2 \end{aligned}

Posted by

infoexpert22

View full answer

Similar Questions

Latest Question