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Let f(x) = {{:({1 + |sin x|}^(a//|sin x|...

Let `f(x) = {{:({1 + |sin x|}^(a//|sin x|)",",-pi//6 lt x lt 0),(b",",x = 0),(e^(tan 2x//tan 3x)",",0 lt x lt pi//6):}` Determine a and b such that f(x) is continuous at x = 0

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The correct Answer is:
`a=2/3,b=e^(2//3)`
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Let y = f(x) be defined in [a, b], then (i) Test of continuity at x = c, a lt c lt b (ii) Test of continuity at x = a (iii) Test of continuity at x = b Case I Test of continuity at x = c, a lt c lt b If y = f(x) be defined at x = c and its value f(c) be equal to limit of f(x) as x rarr c i.e. f(c) = lim_(x rarr c) f(x) or lim_(x rarr c^(-))f(x) = f(c) = lim_(x rarr c^(+)) f(x) or LHL = f(c) = RHL then, y = f(x) is continuous at x = c. Case II Test of continuity at x = a If RHL = f(a) Then, f(x) is said to be continuous at the end point x = a Case III Test of continuity at x = b, if LHL = f(b) Then, f(x) is continuous at right end x = b. If f(x) = {{:(sin x",",x le 0),(tan x",",0 lt x lt 2pi),(cos x",",2pi le x lt 3pi),(3pi",",x = 3pi):} , then f(x) is discontinuous at

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