Let f(x)=⎧⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪⎨⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪⎩(1+|sinx|)a|sinx|,−π6<x<0b,x=0etan2xtan3x,0<x<π6
If f is continuous at x=0, then which of the following option(s) is correct
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Solution
For f to be continuous at x=0, limx→0−f(x)=limx→0+f(x)=f(0)
Now, limx→0−f(x)=limx→0−(1+|sinx|)a|sinx|(1∞form) =limx→0−e(1−sinx−1)⋅a−sinx=ea limx→0+f(x)=limx→0+etan2xtan3x =elimx→0+tan2x2x⋅3xtan3x⋅23=e23 ∴ea=b=e23 ⇒a=23,b=e23