Limit (mathematics)/Limits

Limits of Functions

Basics

What are limits?

Limits are a way to calculate the value that a function approaches. For instance, we could calculate the value of the function f(x) as x approaches 2. Just as easily we can calculate the value of f(x) as x approaches 20, -2, π, 0, or even ∞.

Why would anyone need limits?

There are a number of reasons that someone might want to use limits:

1. To find the values of functions with asymptotes or missing points

2. To calculate the slope of a point in calculus

3. To prove derivatives in calculus

Notation

The notation of a limit function is fairly simple:

\lim _{x\to p}f(x)=L

This says limit (lim) of f(x) as x approaches p is L.

Usually f(x) is substituted with the contents of the function like so:

\lim _{x\to p}x^{2}+2=L

Properties

$\lim _{x\to \alpha }(f(x)+g(x))=\lim _{x\to \alpha }f(x)+\lim _{x\to \alpha }g(x)$

$\lim _{x\to \alpha }(f(x)-g(x))=\lim _{x\to \alpha }f(x)-\lim _{x\to \alpha }g(x)$

$\lim _{x\to \alpha }(f(x)\cdot g(x))=\lim _{x\to \alpha }f(x)\cdot \lim _{x\to \alpha }g(x)$

$\lim _{x\to \alpha }(f(x)/g(x))=\lim _{x\to \alpha }f(x)/\lim _{x\to \alpha }g(x)$

$\lim _{x\to \alpha }f(x)^{g(x)}=\lim _{x\to \alpha }f(x)^{\lim _{x\to \alpha }g(x)}$

Sample Problem Set #1

Let's say we have the function $f(x)=x^{2}$ . If we want to find the limit as x approaches 4, then:

L=\lim _{x\to 4}x^{2}+2

Using two properties of limits:

\lim _{x\to p}x+b=\lim _{x\to p}x+\lim _{x\to p}b

and

\lim _{x\to p}x^{2}=(\lim _{x\to p}x)^{2}

Our problem becomes:

L=(\lim _{x\to 4}x)^{2}+\lim _{x\to 4}2

If we think about the graph of y=b, then we know that the y value never changes. Which means that at any point on that line, we can expect y to be equal to b. So, for any number b:

\lim _{x\to p}b=b

For us, this means that:

\lim _{x\to 4}2=2