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Question
Evaluate the following :
`lim_(x -> 0) [(sqrt(1 - cosx))/x]`
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Solution
Let f(x) = `(sqrt(1 - cosx))/x`
= `(sqrt(1 - cosx))/x xx (sqrt(1 + cosx))/(sqrt(1 + cos x))`
= `(sqrt(1 - cos^2x))/(xsqrt(1 + cosx))`
= `(sqrt(sin^2x))/(xsqrt(1 + cos x))`
= `|sinx|/(xsqrt(1 + cos x)`
Now, |sin x| = `{(sinx, "if" x > 0),(-sinx, "if" x < 0):}`
∴ `lim_(x -> 0^+) "f"(x) = lim_(x -> 0^+) (sqrt(1 - cosx))/x`
= `lim_(x -> 0) sinx/(xsqrt(1 + cosx))`
= `(lim_(x -> 0) (sinx/x))/(lim_(x -> 0) sqrt(1 + cos x))`
= `1/(sqrt(1 + cos 0))`
= `1/sqrt(2)` ...(1)
∴ `lim_(x -> 0^-) "f"(x) = lim_(x -> 0^-) (sqrt(1 - cosx))/x`
= `lim_(x -> 0) (- sinx)/(xsqrt(1 + cosx))`
= `-lim_(x -> 0) sinx/(xsqrt(1 + cos x)`
=`-lim_(x-> 0) ((sinx/x))/sqrt(1 + cos x)`
= `- (lim_(x -> 0) (sinx/x))/(lim_(x -> 0) (sqrt(1 + cos x))`
= `- 1/sqrt(1 + 1)`
= `-1/sqrt(2)` ...(2)
From (1) and (2),
`lim_(x -> 0^+) "f"(x) ≠ lim_(x -> 0^-) "f"(x)`
∴ `lim_(x -> 0) "f"(x) "i.e.", lim_(x -> 0) (sqrt(1 - cos x))/x` does not exist.
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