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If [.] denotes the greatest integer func...

If [.] denotes the greatest integer function then `lim_(x→0)`​`[x^2/(tanx.sinx)]`=

A

0

B

1

C

-1

D

non-existent

Text Solution

Verified by Experts

The correct Answer is:
B
Clearly, `f(x)=(x^2)/(tanx sinx) ` is an even function.
`therefore lim_(xto0^-)f(x)=lim_(xto0^+)f(x)`
In the neighbourhood of `x=0`, we know that
`x gt sinx and x lt tan x`
`rArr (x)/(sin x)gt 1 and (x)/(tan x)lt 1`
Also, `lim_(xto0) (x)/(sinx)=1=lim_(xto0) (x)/(tanx)`
Therefore, in the neighbourhood of `x=0`, we have
`(x^2)/(tanx sin x)gt 1 rArr lim_(xto 0) [(x^2)/(tan x sin x)]=1`.
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