Question

$f\left(x\right)=\frac{logx-log7}{x-7}$ for $x\ne 7$,

Find f(7) if function is continous at x=7

Answer 1

Let x=7+7y then x-7=7y and
log(x) = log[7(1+y)] = log7+log(1+y)
log(x) - log(7) = log(1+y)
= y - (1/2)y^2 + (1/3)y^3 - ...
so
lim x --> 7 [log(x) - log(7)] / [x-7]
= lim y --> 0 [y - (1/2)y^2 + (1/3)y^3 - ..] / [7y]
= 1/7.