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Question

If f(x)=⎪ ⎪⎪ ⎪x22, if 0x12x23x+32, if 1<x2 , Show that f is continuous at x=1.

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Solution

f(x)=⎪ ⎪ ⎪⎪ ⎪ ⎪x22,if 0x12x23x+32,if 1<x2

LHL at x=1,
limx1f(x)=limh0f(1h)=limh0(1h)22=12

RHL at x=1,
limx1+f(x)=limh0f(1+h)=limh0[2(1+h)23(1+h)+32]=23+32=12

Also,
f(1)=122=12

limx1f(x)=limx1+f(x)=f(1)

Hence f(x) is continuous at x=1.

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