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If f(x) = { sin[x] /[x],[x] != 0 ; 0, [x...

If f(x) = { `sin[x] /[x],[x] != 0 ; 0, [x] = 0}` , Where[.] denotes the greatest integer function, then `lim_(x rarr 0) f(x)` is equal to

A

1

B

0

C

`-1`

D

Does not exist

Text Solution

Verified by Experts

The correct Answer is:
d
Given, `f(x) = {{:(sin[x]/[x]",",[x]ne0), (0,[x]=0):},`
`therefore` LHL = `lim_(xto0^(-))(sin[x])/([x])=lim(xto0)(sin[0-h])/([0-h])`
`=lim_(hto0)(-sin[-h])/([-h]) =-1`
RHL `=lim_(xto0)f(x) = lim_(xto0^(+))(sin[x])/([x])`
`=lim_(xto0^(+)) (sin[0+h])/([0+h])=lim_(hto0)(sin[h])/([h])=1`
`therefore LHL ne RHL`
So, limit does not exist.
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