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Question
Select the correct answer from the given alternatives:
If f(x) = `(1 - sqrt(2) sinx)/(pi - 4x), "for" x ≠ pi/4` is continuous at x = `pi/4`, then `"f"(pi/4)` =
Options
`1/sqrt(2)`
`-1/sqrt(2)`
`- 1/4`
`1/4`
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Solution
`1/4`
Explanation:
f(x) is continuous at x = `pi/4`
`therefore "f"(pi/4) = lim_(x -> pi/4) "f"(x)`
`= lim_(x -> pi/4) (1 - sqrt2 sin x)/(pi - 4x)`
`= lim_(x -> pi/4) (sqrt2 (sin x - 1/sqrt2))/(4 (x - pi/4))`
`= sqrt2/4 lim_(x -> pi/4) (sin x - sin pi/4)/(x - pi/4)`
`= sqrt2/4 lim_(x -> pi/4) (2 cos ((x + pi/4)/2) * sin ((x - pi/4)/2))/(x - pi/4)`
`= sqrt2/4 * lim_(x -> pi/4) cos (x/2 + pi/8) * lim_(x -> pi/4) sin ((x - pi/4)/2)/((x - pi/4)/2)`
`= sqrt2/4 * cos (x/8 + pi/8) xx 1 ...[(because x -> pi/4 x - pi/4 -> 0),(therefore (x - pi/4)/2 -> and lim_(theta -> 0) (sin theta)/theta = 1)]`
`= sqrt2/4 xx cos pi/4`
`= 1/4`
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