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!!

  • 2
    $\begingroup$ For $(A)$ and $(B)$, what is the dominant term ? For $(C)$ expand the logarithm. $\endgroup$ Apr 2 '18 at 6:45
  • 1
    $\begingroup$ Your professor probably needs to discuss (with or without proofs) the results like $\lim_{x\to \infty} \log x=\infty,\lim_{x\to\infty}e^x=\infty$ and $\lim_{x\to\infty} x^n/a^x=0$ for $a>1$ before asking you to solve these questions. I hope (s)he has done this otherwise such exercises are pointless. $\endgroup$ Apr 2 '18 at 8:49

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 $1$


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