Since $e^x$ is "transcendental," any way we define it we're going to run into issues with swapping the order of limits. So we'll have to use some polynomial functions to do the dirty work for us. The only polynomials on hand come from the interest formula definition
$$ e^x = \lim_{n\to\infty} \bigg( 1 + \frac{x}{n}\bigg)^n $$
so let's see what we can do with those.

First, show that $1+x < e^x$ for all $x$. This is going to be the "less than" function in an application of the squeeze theorem.

Now the thing to do is take the second-order Taylor polynomial and check what happens. (We're not allowed to talk about "Taylor polynomials," technically, but for $n$ large the binomial expansion of the interest formula is basically the same.)

There are two qualitatively distinct cases. If $x > 0$, then the interest rate formula binomial expansion (i.e. Taylor series) has terms all positive, so the second-order polynomial $1 + x + x^2/2$ is less than the exponential. No good. We'll come back to this in a second.

If $x<0$, then the second order approximation majorizes the exponential function. We know this has to be true because of the alternating series error theorem from Taylor series, but since you're not allowed to use that, let's restrict to $-1<x<0$ and notice that $|x^n|<|x^k|$ if $n>k$, so definitely $|x^n|/n! < |x^k|<k!$. That's the idea in the alternating series error theorem, applied to this particular case.

Back to $x>0$. Let's "cheat" again by observing that there's a whole pencil of parabolas which pass through $(0,1)$ and have $1+x$ as their tangent line at $0$. If we choose one with a large enough quadratic coefficient, we should be set to use the squeeze theorem. Because $1$ is a nice coefficient, let's see if $1+x+x^2$ works. We need $1+x+x^2 > e^x,$ and again I'll indicate the argument with the Taylor series, trusting that you can convert to the binomial approximation.

$$ 1+x+x^2 > 1 + x + \frac{x^2}{2} + \frac{x^3}{3!} + \frac{x^4}{4!}+\cdots$$
is the same as
$$ \frac{1}{2}x^2 > \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots$$
i.e.
$$ \frac{1}{2} > \frac{x}{3!} + \frac{x^2}{4!} + \cdots.$$

We're limiting at zero, so we can let $x$ be small and positive: $|x|<1$, say. In this case,
$$ \frac{1}{2} > \frac{x}{3!} + \frac{x^2}{4!} + \cdots > x\bigg(\frac{1}{3!} + \frac{1}{4!} + \cdots\bigg).$$
That last converges to $e-2$, in fact (this will be easier if you're using binomial approximation I think), so we see that for $0 < x < \frac{1}{2(e-2)}$, we have that $e^x < 1 + x + x^2$.

Bonus: $1+x+x^2 > 1 + x + x^2/2$ for all $x$, so ... for $|x|<\frac{1}{2(e-2)}$,
$$ 1 + x \leq e^x \leq 1 + x + x^2 $$
so
$$ \frac{(1+x)-1}{x} \leq \frac{e^x-1}{x} \leq \frac{(1+x+x^2)-1}{x} $$
and by the squeeze theorem you have the desired result.

**Remarks**.

When using the binomial expansion, you need to show that these inequalities hold for the interest rate formula with *any* (sufficiently large) $n$. It shouldn't be too bad because the coefficients of the binomial series are limiting to the coefficients of the Taylor series.

To make rigorous the use of Taylor series, we need to appeal to the uniform convergence of power series in closed intervals. The general fact is this: Suppose $f_n$ is a sequence of functions and $x$ is in their domain. If $f_n$ converges uniformly, you can swap the limits:
$$\lim_{t\to x}\lim_{n\to\infty} f_n(t) = \lim_{n\to\infty}\lim_{t\to x}f_n(t).$$
This and more is in chapter 7 of Rudin. Results about power series, and the exponential function in particular, are in chapter 8.