Let the function f x = log e 1 + 5 x - log e 1 + α x x if x ≠ 0 10 if x = 0 be continuous at x = 0 . Then α is equal to

Step-by-step solution

f x = ln 1 + 5 x - ln 1 + α x x   x ≠ 0 10   x = 0 For f x to be continuous   lim x → 0 f x = f 0 lim x → 0 ln 1 + 5 x - ln 1 + α x x = 10 Using expansion of ln 1 + x = x - x 2 2 + x 3 3 -   . . . . , we get lim x → 0 5 x - 25 x 2 2 + … … - α x - α 2 x 2 2 + … … x = 10 i.e. 5 - α = 10 ⇒ α = - 5