If f(x) =x(e^([x]+|x|-2))/([x]+|x|), the...

If `f(x) =x(e^([x]+|x|-2))/([x]+|x|)`, then `lim_(xto0)f(x)` is.

A

-1

B

0

C

1

D

non-existent

Text Solution

Verified by Experts

The correct Answer is:

D

We have,
`lim_(xto0^-)f=(x)lim_(xto0^-) x(e^(-1-x)-2)/(-1-x)[because [x]=-1and |x|=-x " for" -1 lt x lt 0]`
`rArr lim_(xto0^-)f(x)=0xx(e^-1-2)/-1=0`
and , ` lim_(xto0^+)f(x)=lim_(xto0)x((e^-x-2)/(x)) [because [x]=0 and |x|=x "for"-1lt x lt 0]`
`rArr lim_(xto0^+)f(x)=lim_(xto0)e^x-2=1-2=-1`
`therefore lim_(xto0)f(x)` does not exist.

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