I'm in Calculus I and am learning about limits at infinity. Here's a problem I have and have been trying to figure out for a long time, to no avail:
For which of the following pairs of functions $f$ and $g$ is $\lim\limits_{x \to\infty} \frac{f(x)}{g(x)}$ infinite?
(A) $f(x) = 3^{x}$ and $g(x) = x^{3}$
(B) $f(x) = 3e^{x}+x^{3}$ and $g(x) = 2e^{x}+x^{2}$
(C) $f(x) = \ln(3x)$ and $g(x) = \ln(2x)$
Professor says that this problem has to be completed without a calculator. I've learned some ways to solve limits at infinity (for example, dividing the numerator and denominator through by something). However, I can't seen to find a way to solve any of these limits algebraically!
Anybody know how to do this type of problem? Thank you!!
Best Answer
For B you Can write $$\frac{3+\frac{x^3}{e^x}}{2+\frac{x^2}{e^x}}$$ And for C write $$\frac{\ln(3)+\ln(x)}{\ln(2)+\ln(x)}$$ The searched Limit is $1$