Continuity Problem - 15th Dec

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!!