Question

Determine f(0) so that the function f(x) defined by f(x) = (4x-1)³ / [sinx/4 log(1+x²/3) becomes continuous at x=0.

Answer 1

Let f(x) = (4^x - 1)³/(sin(x/4) log(1 + x²/3)) for all nonzero x.
(I'll assume the log is base 10; otherwise ignore the ln 10 factor below.)

In order for f to be continuous at x = 0, we need f(0) = lim(x→0) f(x).
So, we need
f(0) = lim(x→0) (4^x - 1)³/(sin(x/4) log(1 + x²/3))
......= lim(x→0) [(4^x - 1)/x]³ / [(sin(x/4)/x) * (log(1 + x²/3)/x²)], dividing numerator and denominator by x³.

Next, observe that
lim(x→0) (4^x - 1)/x
= lim(x→0) (4^x ln 4)/1, by L'Hopital's Rule (0/0)
= ln 4.

lim(x→0) sin(x/4)/x
= lim(x→0) (1/4) cos(x/4)/1, by L'Hopital's Rule (0/0)
= 1/4.

lim(x→0) log(1 + x²/3)/x²
= lim(t→0) log(1 + t/3)/t, letting t = x²
= lim(t→0) [(1/3)/(1 + t/3)] * ln(10) / 1, by L'Hopital's Rule (0/0)
= (1/3) ln 10.

Therefore, we need
f(0) = (ln 4)³ / [(1/4) * (1/3) ln 10] = 12 (ln 4)³ / ln 10.