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Question

Let f(x)={xpsin1x,x00,x=0 then f(x) is continuous but not differential at x = 0 if


A
0<p1
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B
0p<
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C
<p<0
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D
p=0
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Solution

The correct option is B 0<p1
f(x)=xpsin1x,x0 and f(x)=0,x=0
Since at x = 0 , f(x) is a continuous function
 limx0f(x)=f(0)=0limx0xp sin 1x=0p>0
f (x) is differentiable at x = 0, if limx0f(x)f(0)x0 exists
limx0xp sin 1x0x0 exists
limx0xp1sin1x exists
p1>0 or p>1
If p1, then limx0xp1sin(1x) does not exist and at x = 0 f (x) is not differentiable.
for 0<p1 f (x) is a continuous function at x = 0 but not differentiable.

Mathematics

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