How would you evaluate this problem ... f(x) = 2e^4x - 1 and g(x) = 4e^4x + 3.?

Question

Let f(x) = 2e^4x - 1 and g(x) = 4e^4x + 3.
lim.........f(x) .
x=>oo.....g(x)

Answer 1

Do you mean lim (x->inf) f(x) / g(x) ?

Let u=e^x. Then, since u->inf as x->inf, you can rewrite this as

lim(u->inf) (2u^4 - 1) / (4u^4 + 3).

Divide the numerator and denominator by u^4. You get

lim(u->inf) (2 - 1/u^4) / (4 + 3/u^4).

As u->inf, 1/u^4 -> 0, so you're left with 2/4 = 1/2.