Limits

The limit at an x value is what the graph of the function approaches as it approaches that x, regardless of the actual value at that x-coordinate. Do not confuse this with the actual value. You'll see the difference in the examples ahead.

A limit will appear in this form: math $\lim_{x\to c}f(x)=L math We take this to mean the limit of f(x) as x approaches c equals L.

So, for the most basic example, let's assume f(x)=x and c= 5... math $\lim_{x\to 5}x=?? math This one is straightforward. As x gets closer and closer to 5, so does f(x). Don't start thinking they're all this easy.

math $\lim_{x\to 0}\frac{sinx}x=?? math This, seemingly, does not work. If we just tried to plug in 0, we end up with that terror of mathematicians everywhere: division by zero! Instead, let's try something. Pull out a calculator and graph this function. Then, go to Table Settings and change them to start at zero, with a small increment like .1. Then look at the table. If we look at -.3, -.2, then -.1 or on the other side .3, .2, then .1, the values seem to noticeably approach a number: 1. The smaller the increment, the closer to 1 the values you'll see will be. But wait! How can this make sense? math \frac{sin(0)}0 \ can't \ equal \ 1! math Ah, here's where the first sentence above comes in. **It does not matter what the y-value at the x in question is.** All that matters is what value(s) is approached from both sides of that x value.

Wait, both sides? Yes, both sides.

From Wikipedia: math The \ limit \ as: \ x \to x_0^+ \neq \ x \to x_0^-. \\\\\ Therefore, \ the \ limit \ as \ x \to x_0 \ does \ not \ exist. math

Basically, to determine if a limit exists (Yeah, **exists**. We're not even talking about finding out what it is yet.) we must look at a limit from both sides. Find what value is approached both increasing and decreasing towards x. A limit only exists if both of those are equal. For example, above we looked at the limit of that strange function as it went -.3, -.2, -.1 (aka 0-) and on the other side .3, .2, .1 (0+). We then may notice that as x → 0-, it equaled 1. Next, we can see that as x → 0+, it also equaled 1. Therefore, we //could// say: math $\lim_{x\to 0^+}\frac{sinx}x=\lim_{x\to 0^-}\frac{sinx}x math And we will say it, because that statement right above lets us say

math $\lim_{x\to 0}\frac{sinx}x \ exists math

If we look at the Wikipedia graph, it is easily visible that the y's do not approach the same value at x0. Therefore, we can say math $\lim_{x\to x_0}f(x) \ does \ not \ exist math

__How to find a limit__
1. The easy ones: Sometimes, you can just plug in a value and get an answer straight out: math $\lim_{x\to 5}x=5 math math $\lim_{x\to 5}x^3=125 math

2. Simplifying: Sometimes you'll have something like this:

math $\lim_{x\to 5}\frac{x^2-5x}{x-5}=?? math where plugging in gives you a zero on the bottom. First, check if plugging in gives you 0/0. In this case it does, so you have to simplify: math $\lim_{x\to 5}\frac{x(x-5)}{x-5}=$\lim_{x\to 5}x=5 math If plugging in in the first place doesn't give you 0/0, it can't be simplified.

To be continued