When is it sufficient to use logic as proof for an intuitive answer

Question

Say I have the following limit

$$\lim_{x\rightarrow\infty}\frac{3^x}{e^{x-1}}$$

In this case it's simple enough to write it as $\lim_{x\rightarrow\infty}{3^x}{e^{1-x}}$ and then show it approaches infinity, but if there wasn't an easy algebraic solution would it be sufficient to state something along the lines of:

$3^x$ approaches infinity more quickly $e^{x-1}$ because $3$ is larger than $e$ and so will be grow faster with the exponential, and because at each term $e$ is being raised to a smaller exponent.

If not, would it ever be sufficient, and if so, in what context? Am I informally quoting proofs/laws in that statement without even knowing it, or is more work required?

It feels as if I'm saying things that I know to be true without actually substantiating them; it's a line I've been unable to clearly draw for myself ever since I began studying college math.

Answer 1

Note that every proof start by intuition if you don't see the result intuitively it will very difficult because you will never know where to go, in your example it's clear that $\frac{3^x}{e^{x-1}}$ will tend to $\infty$ because the powers of $3$ grows faster than the powers of $e$ because $3>e$ now that you observed this "key" idea and the reasons why $3>e$ this idea works you can write things formally: $$\lim_{x\to\infty} \frac{3^x}{e^{x-1}}=\lim_{x\to\infty}\frac{1}{e}\left(\frac{3}{e}\right)^x\xrightarrow{\color{#a00}{3>e}}\frac{1}{e} \infty=\infty$$ and we have to make sure that we used the reason $3>e$. For me everything starts from intuition and then you try to write what you're thinking more formally and rigorously, this for the question for which the justification is not very "deep" particularly for a question like "prove that:$\cdots$" . Sometimes you can not start from intuition when for examples you try to solve an equation and this case you're not trying to "justify" something but you're looking for the solutions or for the formula let's say for the answer so in the beginning your work is to find the answer and you don't really do things formally and in the end when you found the answer you have to write things formally and rigorously.

Answer 2

hint: $\dfrac{3^x}{e^{x-1}} > \dfrac{3^{x-1}}{e^{x-1}} = \left(\dfrac{3}{e}\right)^{x-1}$

Answer 3

You should rather rewrite $$\frac{3^x}{e^{x-1}}=\frac{e^{x\ln 3}}{e^{x-1}}=e^{x\ln 3-x+1}=e^{x(\ln 3-1)+1}$$ and then argue that $\ln 3-1>0$ (whihc is the case precisely because $3>e$).