Limit (mathematics)/Limits

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:

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

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:

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

Properties

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

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

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

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

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

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:

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

Using two properties of limits:

$\underset{x\to p}{\lim }x+b=\underset{x\to p}{\lim }x+\underset{x\to p}{\lim }b$

and

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

Our problem becomes:

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

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:

$\underset{x\to p}{\lim }b=b$

For us, this means that:

$\underset{x\to 4}{\lim }2=2$

See Also

Wikipedia: Limit of a function