# AoMath: Basic Limit Info

### What Is A Limit?

Basic limit problems look like:
$$\lim_{x \rightarrow 3} \frac{x^2 - 9}{x - 3}$$
The limit above is "The limit as x appoaches 3 of the quantity $$x^2 - 9$$ divided by the quantity $$x - 3$$." More generally, we have:
$$\lim_{x \rightarrow c} f(x)$$or "The limit as x approaches c of the function f of x."
Because of later problems, any author should point out that c and f(x) in the above limit could be "anything" as in $$f(x) = g(x) - x^2 + \ln(x)$$ and $$c = d^2 - 4$$ making the limit:
$$\lim_{x \rightarrow d^2 - 4} g(x) - x^2 + ln(x)$$

$$\lim_{x \rightarrow 3} \frac{x^2 - 9}{x - 3}$$
So what does "approaches" mean in the definition of a limit. If you say x approaches 3, it means you want x to be numbers REALLY close to 3. Then you plug these x's into f(x). This is educated guessing by numberical approximation. Here is our table:
 x = 2x = 2.9x = 2.99x = 2.999x = 2.9999 f(x)55.95.995.9995.9999 x = 4x = 3.1x = 3.01x = 3.001x = 3.0001 f(x)76.16.016.0016.0001

It would seem that f(x) gets closer and closer to 6 as x gets closer and closer to 3. So by "inspection" and educated guessing the limit is 6.