This is the problem that I am trying to prove without bringing up continuity since I won't learn about it for another 4 chapters. I've already proven that $\lim_{x \to c} f(x) = f(c)$ for polynomial functions.
If $f$ is a rational function, then $\lim_{x \to c} f(x) = f(c)$, provided $c$ is in the domain of $f$.
This is my proof:
- Since $f(x)$ is rational function, by definition $f(x) = \frac{p(x)}{q(x)}$ for some polynomial functions $p, q$, such that $q(x) \neq 0$ for all $x$ in the domain of $f$.
- Since it is given that $c$ is in the domain of $f$ we can conclude that $q(c) \neq 0$.
- Since $q$ is a polynomial function, we know that $\lim_{x \to c} q(x) = q(c)$. We also showed that $q(x) \neq 0$, thus it follows that $\lim_{x \to c} q(x) \neq 0$ as well.
- From here $\lim_{x \to c} f(x) = \lim_{x \to c} \frac{p(x)}{q(x)} = \frac{\lim_{x \to c} p(x)} {\lim_{x \to c} q(x)} = \frac{p(c)}{q(c)} = f(c) $
Chat GPT claims that I have a logical inconsistency in step 3, and I kind of get why, but not quite... Do I really need to reason about the neighborhood of $c$ if I know that $\lim_{x \to c} q(x) = q(c)$? Why can't I conclude that $\lim_{x \to c} q(x) \neq 0$?
Note: This is not for a class. I am in my late 30s studying on my own.
Your argument looks fine to me. Please do not rely on ChatGPT for logical arguments. Language models are completely unable of logical reasoning. Unless you ask ChatGPT questions, where the answer can be found on the internet an abundant number of times, ChatGPT will just output something that sounds fitting in the current situation.
ChatGPT has learned, that saying "You made a mistake in Step 3" sounds like a plausible answer to any type of question about mathematical reasoning. It did not actually check your arguments.