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Question: If \(f(x) = \left\{ \begin{matrix} x,\text{when}x > 1 \\ x^{2},\text{when}x < 1 \end{matrix} \right....

If f(x)={x,whenx>1x2,whenx<1 ,thenlim⁡x→1f(x)=f(x) = \left\{ \begin{matrix} x,\text{when}x > 1 \\ x^{2},\text{when}x < 1 \end{matrix} \right.\ ,\text{then}\lim_{x \rightarrow 1}f(x) =

A

x2x^{2}

B

x

C

– 1

D

1

Answer

1

Explanation

Solution

To find L.H.L. at x=1.x = 1. i.e.,

lim⁡x→1−f(x)=lim⁡h→0f(1−h)\lim_{x \rightarrow 1^{-}}f(x) = \lim_{h \rightarrow 0}f(1 - h) = lim⁡h→0(1−h)2\lim_{h \rightarrow 0}(1 - h)^{2} = lim⁡h→0(1+h2−2h)\lim_{h \rightarrow 0}(1 + h^{2} - 2h) = 1

i.e., lim⁡x→1−f(x)=1\lim_{x \rightarrow 1^{-}}f(x) = 1 ….(i)

Now find R.H.L. at x = 1 i.e., lim⁡x→1+f(x)=lim⁡h→0f(1+h)\lim_{x \rightarrow 1^{+}}f(x) = \lim_{h \rightarrow 0}f(1 + h) = 1 i.e., lim⁡x→1+f(x)=1\lim_{x \rightarrow 1^{+}}f(x) = 1 …..(ii)

From (i) and (ii), L.H.L. = R.H.L. ⇒ lim⁡x→1f(x)=1\lim_{x \rightarrow 1}f(x) = 1.