Question

f(x) = (a^([x] + x) - 1)/([x] + x) for x != 0

f(x) = log a for x = 0

Then f(x) is continuous at x = 0

a)" "True

b)" "False

c)" "Continuous if loga = 1 - 1/a

d)" "Cant Say

Solution

To check continuity of the function, we have to check for left and right hand limits.

So for f(0-) = (a^(-1 + x) - 1)/(-1 + x)

=> f(0-h) = Lim_(h->0)(a^(-h)/a - 1)/(-h - 1) = 1 - 1/a

For f(0+) = (a^(0+x) - 1)/(0+x)

=> f(0+h) = Lim_(h->0)(a^h - 1)/h

Applying L'Hopital rule

=> f(0+h) = Lim_(h->0)(a^h - 1)/h = loga

Also f(0) = loga

For function to be continuous, f(0) = f(0-h) = f(0+h)

Hence, loga = 1 - 1/a