A function's bounds The limit of w = f(z) as z-z0 is a number l such that we can make f(z)-l as tiny as we wish by making z-z0 sufficiently small. In **other circumstances**, such as w = z2-z, the limit is simply f (z0). This function's limit as z-i, for example, is f(i) = i2-i = -1-i. The function has no limit at all if there is no number l such that f(z) can be made as small as we please by making z-z0 small.

A function's limit Assume that f is a real-valued function and that c is a real number. Intuitively, the phrase suggests that by bringing x near to c, f (x) may be made to be as close to L as required. In such scenario, the above equation may be written as "the limit of f of x as x approaches c is L.".

The limit of a function can take on **many different values**. For example, let f(x) = x - 1/x. What is the limit of this function as x approaches 0?

It's not hard to see that if we try to solve this problem directly, we'll get awfully lost in **all those negative signs**! Instead, we can use the fact that the limit of a function is the same as its value at zero, which in **this case** is just 1. So, our solution is that the limit of f as x approaches 0 is 1.

Now, suppose we wanted to find the limit of **another function**, for example g(x) = f(x) - x. It's not too difficult to see that if we try to solve this problem directly, we'll again get stuck with negative signs! But again, we can use the fact that the limit of a function is the same as its value at zero, which in this case is just 1. So, the limit of g as x approaches 0 is 1 too!

A right arrow (-) is occasionally used to express the fact that a function f approaches the limit L as x **approaches c**, as in f (x)-L as x-c. Assume that f is a real-valued function and that c is a real number. Mathematically, this means that if no value of x makes f (x) equal to L, then there must be at least one value of x such that f (x) < L.

A limit is the value that a function approaches as its inputs come closer and closer to a certain number. The concept of a limit lies at the heart of all calculus. Limits play an important role in **many fields** of mathematics, particularly in analysis.

The limit of f(x) as x approaches **an equals L**" implies that we can get f(x) as near to L as we wish by bringing x close enough to a. Thus, there are many numbers close to but less than or equal to L such that their negatives are also close to L.

Since -5 is very close to 5, we know that there are positive integers close to but less than or equal to 5 such that their negatives are also close to 5. For example, -3 is closer to 5 than -5 is to 5; therefore, -3 is greater than 0 and less than or equal to 5.

Now let's try **another number**: -100. -100 is much more distant from 100 than 5 is from 5, so we should expect that there are fewer negative integers close to -100 than there are close to 100. In fact, there are two negatives close to -100: -101 and -99.

This means that if we bring x close enough to **a, then f(x**) will be close to L for at least one value of x close to a. Since we can make L as close to a as we want, this shows that there are always at least two values of x close to **a such that f(x**) is close to a.

A function's value is the actual computation performed at a given moment. The limit is basically the value for sites that are "arbitrarily near" to each other. For example, let's say I give you a function and a limit; can you find the limit of the function? You could probably do this by trying out values in **close proximity** to each other and seeing which one gets closer to the limit.

The problem is that there isn't any clear way to determine how close together two numbers have to be before they're considered "close enough" to use as limits. In fact, there are an infinite number of ways that two real numbers can be arranged. With functions, though, we usually know what the function is going to look like when it takes on **its limit**: it will always be some sort of **arithmetic or geometric mean**. So if we see that some number is closer to the function's limit than another, then we can conclude that the first number is the limit.

Here's an example. Let f be the function that gives back the square of its argument (x). I'll also say that the limit exists and is equal to 3.