What are some examples of non standard limits

Question

Once we accept that polynomials are continuous then there is no need to resort to $\epsilon$-$\delta$ methods to prove limits of polynomials except for educational purposes. Yet many problems and examples in textbooks ask students to use $\epsilon$-$\delta$ methods to prove limits of linear and quadratic functions, and students might feel that they are wasting time on 'toy' problems. What are some examples of functions whose limits must be proved using $\epsilon$ -$\delta$ methods, the proofs of which are accessible to undergraduates?

Answer 1

You're missing the point. It's not a technique; it's a definition. In math, you need to rigorously define all the concepts you discuss, and the epsilon-delta "method" is the way the limit is defined. Being able to work it out for polynomials, or other elementary functions such as trig functions, exponents, logarithms, etc., or more advanced functions, or whatever pathological case you could come up with, comes after that. You can't work something out if you don't first know what, exactly, it is that you're working out.

Answer 2

Every function limit must be proved using "$\delta - \epsilon $ methods", at least indirectly like though the definition of continuity, since this is the definition of a limit. I find calling this "method" quite absurd, by the way.

If you want good examples, put some trigonometric functions. Then, you can use functions defined differently on rationals and irrationals, or on different intervals. That should make them get it. Afterwards, there are functions continuons everywhere and derivable nowhere, you can let them prove the continuity.

Following comment : Apparently, your student has been taught "maths with the hands". You should explain to him that maths are made only of axioms, definitions and theorems you derive thanks to these axioms. Every property or theorem concerning limits bases itself on the definition of limit, otherwise it is talking about something else. This is what I called referring indirectly to the definitions.

There are many cases which are not practically covered by l'hospital rule, the squeezing theorem, and so on. Those apply only in a restrictive context. Example : prove that $ \lim\limits_{x \rightarrow + \infty } \ln(x) = + \infty $ knowing only about the limits of polynomials. The easier way is refer directly to the definition.

Personally, I have never been taught this as a "method", but as a definition. This teaching of "$\delta - \epsilon$ method" is not applied everywhere.

Answer 3

I think you want examples where more than the basic algebraic properties of polynomials and roots are needed.

(1). $\lim_{x\to \infty}(1+y/x)^x=e^y.$

(2). If $f(0)=0$ and $f(x)=e^{-1/x}$ for real $x\ne 0$ then for every $j>0$ the $j$th (real) derivative of $f(x)$ is $0$ at $x=0.$

(3). If $a_n\geq 0$ then $\sum_{n=1}^{\infty}a_n<\infty \iff \prod_{n=1}^{\infty}(1+a_n)<\infty,$ and its companion: If $0\leq a_n<1$ then $\sum_{n=1}^{\infty}a_n<\infty\iff \prod_{n=1}^{\infty}(1-a_n)>0.$... From this we can show (Euler) $\sum_{p\in \Pi}(1/p)=\infty,$ where $\Pi$ is the set of primes.

(4). If $D\subset \mathbb C$ and $(f_n:D \to \mathbb C)_{n\in \mathbb N}$ is a sequence of continuous functions, converging uniformly to $f:D\to \mathbb C$, then $f$ is continuous.

General results, like (3) and (4), can be then shown to be applicable to a wide variety of special cases.