Suppose f and g are continuous functions such that
g(4) = 5
and
lim
x → 4
[3f(x) + f(x)g(x)] = 32.
Find
f(4).
2 Answers

Since f and g are continuous, we have that
lim f(x) = f(a)
x → a
lim g(x) = g(a)
x → a
for all real numbers a in R.
Also, since f and g are continuous, so is 3f + fg so that
lim [3f(x) + f(x)g(x)] = [3f(a) + f(a)g(a)]
x → a
for all real numbers a in R.
In particular,
lim [3f(x) + f(x)g(x)] = [3f(4) + f(4)g(4)]
x → 4
Since we are given that
lim [3f(x) + f(x)g(x)] = 32
x → 4
and g(4) = 5, substituting these into the equation above, we have
32 = [3f(4) + f(4)(5)]
32 = 8f(4)
4 = f(4).
Thus f(4) = 4.

Thank you!!