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Question

# The function f(x)=1x−2e2x−1,x≠0 is continuous at x=0, then

A
f(x) is differentiable at x=0
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B
f(0)=1
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C
f(x) is not differentiable at x=0
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D
f(0)=13
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Solution

## The correct option is D f′(0)=−13limh→0+f(x)=limh→0+f(0+h)=limh→0[10+h−2e2(0+h)−1] =limh→0[1h−2e2h−1] =limh→0e2h−1−2hh(e2h−1)=1 Similarly, limx→0−f(x)=limh→0e−2h−1+2h−h(e−2h−1)=1 Hence f(0)=1 Rf′(0)=limh→0f(0+h)−f(0)h=limh→0{10+h−2e2(0+h)−1}−1h =limh→01h−2e2h−1−1h ==limh→0e2h−1−2h−h(e2h−1)h2(e2h−1)=−13 Similarly, Lf′(0)=−13 ⇒f(x) is differentiable at x=0 and f′(0)=−13

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