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let [x] denote the greatest integer less...

let [x] denote the greatest integer less than or equal to x.
Then `underset(xto0)lim(tan(pisin^2x)+(absx-sin(x[x]))^2)/x^2`

A

does not exist

B

equals 0

C

equals `pi+1`

D

equals `pi`

Text Solution

Verified by Experts

The correct Answer is:
A
`lim_(xto0) ((tan(pisin^2x))/x^2 + ((|x|-sin (x[x]))^2)/x^2)`
At x=0
LHL `lim_(x to 0^(-)) ((tan(pisin^2 x))/x^2 +((-x-sin(-x))^2)/x^2)=pi`
RHL : `lim_(xto0^+)((tan(pi sin^2 x))/x^2 + ((x-0)^2)/x^2)=pi+1`
`therefore` LHL `ne` RHL hence limit doesn't exist
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