Question

evalute the limit of (viii)

Answer 1

See answer in above image

Answer 2

lim h->0 loga(x+h) - logax/h. ......(1)

we know that
d[f(x)]/dx = lim h->0 f(x+h) - f(x)/h. ...(2)

on comparing eqn (1) and (2) we get

f(x) = loga(x)

so
f'(x) = d[loga(x)]/dx
= 1/x loge(a) (by using the property of log)
:')