Limit (mathematics)/Limits

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:
 * $$ \lim_{x \to 4}2 = 2$$