Let `f(x)={(1-sin^3x)/(3cos^2x)\ \ \ ,\ \ \ if\ xpi/2` . If `f(x)` is continuous at `x=pi/2` , find `a` and `b` .

Explanation

If a function is continuous at `x=c` then its limit exist at a and is equal to `f(c)`



`lim _{x -> frac{pi}{2}} f(x)=lim _{x -> frac{pi}{2}} frac{(1-sin ^{3} x)}{3 cos ^{2} x}=lim _{x -> frac{pi^{-}}{2}} frac{(1-sin x)(1+sin ^{2} x+sin x)}{3(1-sin x)(1+sin x)}`




`lim _{x ->-frac{pi}{2}-} f(x)=lim _{x -> frac{pi}{2}} frac{1+sin ^{2} x+sin x}{3(1+sin x)}=frac{3}{3 times 2}=frac{3}{2}`



so Left side limit is `frac{3}{2}` So a `={f}(frac{pi}{2})=frac{3}{2}`


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