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

Let ` f(x) = {{:({1+|sin x|}^(a//|sin x|)", " pi/6 lt x lt 0),(" b, "

1.	Let ` f(x) = {{:({1+\|sin x\|}^(a//\|sin x\|)", " pi/6 lt x lt 0),(" b, " x = 0 ),(e^(tan 2x//tan 3x) ", "0ltx ltpi/6):}` Determine a and b such that f(x) is continous at x = 0.
Answer» Correct Answer - `a=2/3,b=e^(2//3)` ` f(x) = {{:({1+\|sin x\|}^(a//\|sin x\|) " , "pi//6 lt x lt 0),(" , " x = 0),(e^(tan 2x//tan 3x)" , "0 lt x lt pi //6):}` Since, f(x) is continuous at x = 0 . `:. ` RHL (at x = 0) = LHL (at x = 0) = f(0) ` rArr underset( h to 0) lim e^(tan 2h//tan 3h) = underset( h to 0) lim { 1+ \| sin h\| } ^(a//\| sin h\|) = b` ` rArr" " e^(23) = e^(a) = b` ` :." " a = 2//3` ` and" " b = e^(2//3) `