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Question
`lim_(x -> 0) ("cosec" x - cot x)/x` is equal to ______.
Options
`-1/2`
1
`1/2`
– 1
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Solution
`lim_(x -> 0) ("cosec" x - cot x)/x` is equal to `1/2`.
Explanation:
Given `lim_(x -> 0) ("cosec" x - cot x)/x`
= `lim_(x -> 0) (1/sinx - cosx/sinx)/x`
= `lim_(x -> 0) (1 - cos x)/(x sin x)`
= `(2 sin^2 x/2)/(x * 2 sin x/2 cos x/2)` ......`[because sin 2x = 2 sin x cos x]`
= `lim_(x -> 0) (sin x/2)/(x cos x/2)`
= `lim_(x -> 0) (tan x/2)/x`
= `lim_(x -> 0) (tan x/2)/(2 xx x/2)`
= `1/2 xx 1`
= `1/2` ......`[because lim_(x -> 0) tanx/2 = 1]`
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