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If the function `f(x) {((1 - cos 4x)/(8x^(2))",",x !=0),(" k,",x = 0):}`is continuous at x = 0 then k = ?A. 1B. 2C. `(1)/(2)`D. `(-1)/(2)` |
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Answer» Correct Answer - C `underset(x rarr 0)(lim) f(x) = underset(h rarr 0)(lim) f(0 +h) = underset(h rarr 0)(lim) (1 - cos 4h)/(8h^(2)) = underset(h rarr 0)(lim) (2 sin ^(2) 2h)/(8h^(2))` `= (1)/(2) underset(h rarr 0)(lim) ((sin 2h)/(2h))^(2) = ((1)/(2) xx 1^(2)) =(1)/(2)` `underset(x rarr 0)(lim) f(x) = underset(h rarr0)(lim) f(0 -h) = underset(h rarr 0)(lim) (1 - cos 4 (-h))/(8(-h)^(2)) = underset(h rarr0)(lim) ((1 - cos 4h))/(8h^(2)) = (1)/(2)` `:. underset(x rarr 0^(+))(lim) f(x) = (1)/(2)` For continuity, we must have `f(0) = (1)/(2)` |
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