I get confused everytime between finding a limit and proving that something is a limit. If I calculate the limit of a function $f(x)$ at $a$ and say, I get $\lim\limits_{x \to a} f(x) =L$ where $a\in [-∞,∞]$ then is it not the same as showing that $L$ is the limit of $f$ at $a$?
However, whenever I read answers here on this site or on other math sites, I notice that when they're about to calculate a limit of a function or some weird expression, they start off by saying, "assuming the limit exists" or "assuming the expression converges" and then proceed to calculate. What's that about? I don't get it, why do you have to assume that it exists or not? If you do find a limit then surely the thing has a limit, otherwise what does it even mean to find a limit?
My Questions:
- Can someone give me a clear definition of what it means to find or compute vs what it means to prove. And maybe explain the difference through an example?
- Explanation of the phrase, "assuming an expression converges" (Note: not just a function but any expression in some variables, what
does it mean for it to converge in simple English)?- What is the logic behind this? Why do we do this? What logical errors would we encounter otherwise?
Edit: I'm updating my question and including an example of what I'm trying to ask.
Say, $f(n)= \dfrac{3n^2+1}{2n^2-89}$ then, I can compute the limit:
$$\lim\limits_{n \to ∞} f(n) = \lim\limits_{n \to ∞} \dfrac{3n^2+1}{2n^2 -89} = \lim\limits_{n\to ∞} \dfrac{3 + \frac{1}{n^2}}{2- \frac{89}{n^2}} = \dfrac{\lim\limits_{n \to ∞} \left(3 + \frac{1}{n^2}\right)}{\lim\limits_{n \to ∞}\left(2- \frac{89}{n^2}\right)}= \dfrac{3}{2}$$
Here, I can justify how I can go from one step to the other. For instance, I can go from step 2 to step 3 because as $ n \to ∞, n ≠0$, I can make this more rigourous but I'll not for brevity. Now, why do I have to prove that $3/2$ is really the limit of $f(n)$ when $n \to ∞$.
In the example below in the comments, Mr. Dave give $x_n = 2+4+8+ \ldots + 2^n$ and claim (without actually computing) $L$ to be the limit. Since, he has not computed and found $L$ to be real number. He cannot use it around like a real number as well, so the manipulation that follows in his example doesn't make any sense. Whereas I have found my limit (and it is a real number) through proper justification and reasoning while going from one step to the other. Why do I still need to prove that $3/2$ is the limit or that $f(n)$ converges? What is the logic/reasoning behind it?
Best Answer
When your method for calculating a limit is valid, i.e. the method is justified as a consequence of the definition of a limit (for example, the definition of the limit of a sequence of real numbers implies that whenever $\lim x_n = x$ and $\lim y_n = y$ then $\lim x_n + y_n = x + y$ for any two sequences $(x_n),(y_n)\in \mathbb{R}^\mathbb{N}$ and reals $x,y\in\mathbb{R}$), obviously finding the limit is proving such a limit exists. The converse of course is not true.
A typical example of limits we usually can prove to exist but often we do so without computing explicitely their values are the limits of real sums of the form $\lim_{n\to\infty} \sum_{k=0}^n c_k$ for $c_k \in\mathbb{R}$. In this case saying the limit exist amounts to say the series converges, i.e. $\sum_{k=0}^\infty c_k = x$ for $x\in\mathbb{R}$, but calculating $x$ is usually harder than (dis-)proving this convergence.
When you compute a limit of a real function through the "typical" means, apparently without knowing it, you are using a set of rules derived from the definition of a limit as well as different theorems, so for example you might say $\lim_{t\to\infty} \frac{1}{t} = 0$ because, by the monotone convergence theorem, the limit must be equal to its infimum, namely $0$.
The definition of convergence, in English, can be found in any dictionary which, as far as I know, this forum is not. To build an intuition of the meaning of mathematical convergence you probably should start by considering more thouroughly the definition of limit of a sequence or of a function on a metric space. In plain words, and for example, the limit of a real function $f$ at a point $x_0 \in \mathbb{R}$, when the limit exists, is this value $L\in\mathbb{R}$ that, the closer $x$ gets to $x_0$, the closer $f(x)$ gets to $L$.
You might wonder "Why do you specify when the limit exists?". Well, you cannot talk of the limit $L\in\mathbb{R}$ of $f$ at $x_0$ if, as a matter of fact, there is no $L\in\mathbb{R}$ that $f(x)$ approches as $x$ approaches $x_0$. For example, we cannot properly talk of the limit of $\frac{1}{t-1}$ at $1$ because there is no real number that $\frac{1}{t-1}$ approaches as $t$ approaches $1$. We conventionally note $\lim_{t\to 1}\frac{1}{t-1} = \infty$ not to say $\infty$ is a real number to which $\frac{1}{t-1}$ converges as it approaches $1$, but to say precisely the opposite: that $\frac{1}{t-1}$ diverges as it approaches $1$ and that, concretely, as $\frac{1}{t-1}$ aproaches this value it becomes unlimitedly large.