What is a Limit?

This limit thing keeps coming up in my calculus textbook. What is it?

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For many functions there are undefined values. For example in the function $$f(x) = \frac{x^2 – 1}{x – 1}$$ there is no value of $f(x)$ for $x = 1$, since that involves a division by 0.
In this instance though, as x gets closer and closer to 1, $f(x)$ gets closer and closer to 2, so we define that to be the limit.

In calculus limits come up when getting the slope of a linear function as I’m sure you’ve seen, the problem is that getting the slope at a certain point, using the slope formula we all know, involves dividing by zero, so we use limits to get the answer. More specifically we create a formula for the slope between the point $(x, f(x))$ and the point $(x+h, f(x+h))$ and find the slope as $h \to 0$.

Limits as $x \to \infty$ are also used often, since even if a function is well defined for all $x$ it is still impossible to simply plug in infinite and work it out, in this way it can be used for approximation purposes. Calculating the runtime of computer programs on large inputs is often done by deriving a formula for the run-time given input of a certain length, then finding the limit as $x \to \infty$

What others have provided is a pretty good intuitive/more general definition of a limit. If you want something a bit more technical and rigorous, have a look at the (ε, δ)-definition of limit, which is the one used in mathematical analysis.

A limit is the term used when you examine a function as it approaches some value. For example, the function $f(x) = \frac1x$. You can take a limit of this function as $x$ approaches infinity and as $x$ approaches $0$ and see what occurs, and get an idea of the behaviour of the function in these regions even though the function cannot be defined at the exact values you have taken limits of. ($\frac10$ is undefined, and $\frac{1}{\infty}$ cannot be done without adding infinity to the number system – see the hyperreals for this)

We write
$$\lim_{x→a} f(x) = L$$
and say

“the limit of $f(x)$, as $x$ approaches $a$, equals $L$”

if we can make the values of $f(x)$ arbitrary close to $L$ (as close to $L$ as we like) by taking $x$ to be sufficiently close to $a$ (on either side of $a$ but not equal to $a$.

More formally,

Let $f$ be a function defined on some open interval that contains the number $a$, except possibly at $a$ itself. Then we say that the limit of $f(x)$ as $x$ approaches $a$ is $L$, and we write
$$\lim_{x→a} f(x) = L$$
if for every number $\epsilon > 0$ there is a number $\delta > 0$ such that $|f(x) – L| < \epsilon$ whenever $0 < |x – a| < \delta$.

There are two slightly different types of limits (at least, in one real variable, as in single-variable calculus): limits as a variable goes to positive or negative infinity, and limits as a variable goes to some real value.

The limit of $f(x)$ as $x$ goes to infinity (or as $x$ goes to negative infinity) is asking what happens to $f(x)$ at the right (left) end of its graph. Does $f(x)$ get arbitrarily close to a single real number as $x$ increases (or decreases) without bound–that is, as $x$ gets really big and positive (or really big and negative)?

The limit of $f(x)$ as $x$ goes to some real value, say $c$, is asking what happens to $f(x)$ near, but not at, $x = c$. Does $f(x)$ get arbitrarily close to a single real number as $x$ gets arbitrarily close to $c$?

“Arbitrarily close” is used somewhat informally above, but these ideas can be translated into more rigorous definitions (as in other people’s answers) by reformulating “arbitrarily close” in terms of inequalities with $\varepsilon$; and/or $\delta$. Also, while a limit is said to not exist if there isn’t the single real number as described above, it is somewhat common to talk about the limit being infinity (or negative infinity) if the limit does not exist, but the values of $f(x)$ increase without bound (or decrease without bound).

If you were an ant crawling along a curve, at a really small (almost zero) distance away from the point you’re taking the limit of, where are you?

A rigorous mathematical treatment follows:

We say the limit of $f(x)$ as $x$ approaches $c$ is $\tau$ iff for all $\varepsilon$ in $\mathbb R^+$ there exists delta in $\mathbb R^+$ such that $|x-c|< \delta$ implies $|f(x)-\tau| < \varepsilon$.

Adding another “famous” heuristic explanation would be Zeno’s paradox(es), which is often the first thing mentioned in many Calculus classes when discussing limits.

Before an object can travel a given distance d, it must travel a distance call it d/2. In order to travel d/2, it must travel d/4, etc. This appears to go on forever, and one can reason that the distance d cannot be traveled, as there are infinitely many half steps. But from real world experience, this does not hold up as it is obviously possible to travel and actually arrive at a certain destination.

A limit is simply a mathematical tool used to extend the domain of definition of a function in a way that circumvents circular logic (i.e. assuming the antecedent). For instance, let f be a function defined on a domain X, and suppose that the point p is not contained in X, but also that we would like to examine the behavior of f at or near the point p. For the sake of simplicity, assume that we are only interested in the value of f. One may jump the gun and say that the value of f at p is f(p), however this may not necessarily be true. Moreover, we have implicitly assumed that the domain of definition is extendable to p, which is the assumption of the antecedent. In other words, we have assumed the value of f at p is f(p) to prove that the value of f at p is f(p), which is in general a logical fallacy. Thus, we introduce the notion of limit to recursively extend the domain of definition of f.

Example:
Suppose the desired point p=1 and define f as
$$f\left( x \right) := \frac{{{x^2} – 1}}{{x – 1}}$$
Now, x=1 is not in the domain of definition of f since the denominator is 0 at x=1. So, the function f looks like a line with a hole in it at x=1. But, having noticed that we can factor the numerator and simplify, we have for x≠1
$$f\left( x \right) = x + 1$$
Evaluating the limit then gives
$$\mathop {\lim }\limits_{x \to 1} f\left( x \right) = \mathop {\lim }\limits_{x \to 1} \left( {x + 1} \right) = 2$$
Assuming that we want the to extend the function f to a function g which is continuous on the extended domain, we simply define g piecewise as

g(x)=f(x) if x≠1, and g(x)=2 if x=1

Thus g(x)=x+1.

This example is meant to explain not only what a limit is, but also why we use them.

Conceptually, to me at least, a limit is a way approximate something, by realizing the long run behavior is predictable and more important then the short term behavior.

Let’s say your baby sitting a rambunctious kid. You could frustrate yourself trying to quite him, or just realize, in the limit he will tire himself out.

A lot of systems seem complicated in the short term, but are simple when viewed in the long term.

The limit is way to take advantage of that, and relate a complex system to a simpler approximation.

The way I understood it when I first learnt it, limit is the characteristic of a function at a point. If it has limit $L$ at $x=a$, then it behaves more and more like $L$ as it comes close at point $a$.

If it doesn’t have limit, then it shows many characteristics (not just one) or doesn’t even exist.

I always understood the $\epsilon-\delta$ in terms of scissors. With $\delta$ being the distance between two handles and $\epsilon$ being the distance between the tip of blades. Let’s say one tip of blade has the limit as other blade as you make $\delta$ small and small the $\epsilon$ becomes smaller and smaller and the tip of the blades approach close and close.

But how close can you make without making them equal? You can make them as close you want!

A limit’s technical definition is: “a point or level beyond which something does not or may not extend or pass.”

This is just a description, yet it applies to Calculus as well. A limit is almost like the symbol infinity, in that it acts like a number but it is not quite.

You will see later in your Calculus class how the limit will help to find many things that are very important.