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If f(x)={{:(,x^(2)sin((1)/(x)),x ne 0),(...

If `f(x)={{:(,x^(2)sin((1)/(x)),x ne 0),(,0, x=0):}`, then

A

`f(0+0)=1`

B

`f(0-0)=1`

C

f(x) is continuous at x = 0

D

None of the above

Text Solution

Verified by Experts

The correct Answer is:
C
`lim_(x to 0^(+))f(x)=lim_(x to 0^(+)) x^(2) "sin"(1)/(x)`
`therefore lim_(x to 0^(+)) f(x)=0xx("an oscillating value i.e. between"- 1 and 1)=0`
`[ because lim_(x to 0^(+))"sin"(1)/(x)="an oscillating value i.e. between -1 and 1 as -1" le sin x le 1]`
Similarly, `lim_(x to 0^(-))f(x)=0`
Hence, f(x) is continuous at x = 0.
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