I am currently reading "Calculus a Rigorous First Course" by Daniel J. Velleman. I am on the exercise set of section 2.4 problem 29, and it states the following.
"Suppose that $f$ and $g$ are functions that agree on all values except one. In other words, there is some number $c$ such that for all $x \neq c$, $f(x) = g(x)$, but $f(c) \neq g(c) $. Show that for every number a, $\lim_{x \rightarrow a}f(x)=\lim_{x \rightarrow a}g(x)$, where we interpret this equation to mean that either both limtis are defined and they are equal, or both limits are undefined."
My attempt:
(Let $\lim_{x \rightarrow a}f(x)=\lim_{x \rightarrow a}g(x)=L$)
Proof. Supppose $\epsilon > 0$. Let $\delta=\min(\delta_1, \delta_2)$, such that if $0<|x-a|<\delta_1$ then $|f(x) – L| < \epsilon/2$. Similarly, if $0<|x-a|<\delta_2$ then $|g(x)-L|<\epsilon/2$. Now, suppose $0<|x-a|<\delta$. Because $\delta \leq \delta_1$ then $0<|x-a|<\delta_1$ which means that $|f(x)-L|< \epsilon /2$. Similarly, because $\delta \leq \delta_2$ then $0<|x-a|<\delta_2$, so $|g(x)-L|< \epsilon/2$. Therefore,
$$|f(x) – L+g(x)-L)| \leq |f(x)-L| + |g(x)-L| < \epsilon/2 + \epsilon/2 = \epsilon. $$
My doubt
I do not see how my proof has anything to do with the condition $f(c) \neq g(c)$. Also, I would like to add an additional note left on the problem, which states
"(Note: We could state this result more informally by saying the value of the limit $\lim_{x \rightarrow a}f(x)$ will not be affected if we change the value of $f(c)$, for any number $c$. This may seem paradoxical: it suggests that none of the values of the function $f$ are relevant to the value of $\lim_{x \rightarrow a}f(x)!)$"
How is the factorial of $f(x)$ relevant? Or this some sort of typo?
Best Answer
You proved that$$\lim_{x\to a}f(x)=L\wedge\lim_{x\to a}g(x)=L\implies\lim_{x\to a}f(x)+g(x)=2L.$$But that is not what you were supposed to have proved.
Suppose that $\lim_{x\to a}f(x)=L$. If it turns out that $a=c$, then this is exactly the same thing as proving that $\lim_{x\to a}g(x)=L$, since the value that $f$ and $g$ take at $c$ doesn't matter and $f(x)= g(x)$ when $x\ne c \wedge c=a$.
And if it turns out that $c\ne a$, take $\varepsilon>0$ and take $\delta>0$ such that$$0<|x-a|<\delta\implies|f(x)-L|<\varepsilon.$$Now, let $\delta'=\min\{\delta,|c-a|\}$. Then $0 <|x-a|<\delta'$ implies two things:
And therefore $|g(x)-L|=|f(x)-L|<\varepsilon$.
Now, if the limit $\lim_{x\to a}f(x)$ doesn't exist, then the limit $\lim_{x\to a}g(x)$ cannot exist, since otherwise, by the previous argument, if it did, then them limit of $f$ at $a$ would also exist.