Question

If f is continuous at x = 0, then find f (0). 

Where f(x) = `(3^"sin x" - 1)^2/("x" . "log" ("x" + 1)) , "x" ≠ 0`

Answer 1

Given f is continuous at x = O 

∴ f (0) = `lim_(x->0) "f(x)"`

∴ f (0) = `lim_(x->0) [(3^"sin x" - 1)^2/("x" . "log" ("x" + 1))]`

`= lim_(x->0) [(3^"sin x" - 1)^2/("sin"^2 "x")  × ("sin"^2"x")/("x" . "log" ("x" + 1))]`

∴ f (0) = `lim_(x->0) ((3^"sin x" - 1)/"sin x")^2 . (("sin"^2 "x")/"x"^2)/(("x log" (1 + "x"))/"x"^2)`

∴ f (0) = `lim_(x->0) ((3^"sin x" - 1)/"sin x")^2 . (lim_(x->0)(("sin x")/"x")^2)/(lim_(x->0) [("log" (1 + "x"))/"x"])`

`= "(log 3)"^2 . (1)^2/"log e" = ("log" 3)^2 xx 1/1`

∴ f (0) = `("log 3")^2`