For an assignment on Taylor expansion we are given the following limit to determine:
$\lim_{x \to 0} \frac{\frac{2}{1 + e^x} - 1 + \frac{x}{2}}{x^3}$
From a previous subproblem of the assignment I have already expressed a Taylor polynomial of the third degree as:
$p_3(x) = 1 - \frac{1}{2} x + \frac{1}{24} x^3$
My solution is, therefore, to write the expression with the additional term $\mathcal{O}(x^4)$ so that:
$\lim_{x \to 0} \frac{\frac{2}{1 + e^x} - 1 + \frac{x}{2}}{x^3} = \lim_{x \to 0} \frac{1 - \frac{1}{2} x + \frac{1}{24} x^3 + \mathcal{O}(x^4) - 1 + \frac{x}{2}}{x^3} = \lim_{x \to 0} \frac{\frac{1}{24} x^3 + \mathcal{O}(x^4)}{x^3} = \lim_{x \to 0} \frac{1}{24} \mathcal{O}(x) = \frac{1}{24}$
Any additional terms in the expansion will contain $x$ and thus approach zero and not contribute to the value of the limit.
A couple of questions:
- Does this look like a valid solution to the problem?
- What are some conceptual ways to look at this, if I want to fully understand why I can substitute an exact expression with an approximation? To me it still doesn't sit right that this is allowed.