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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 = 2
x = 2.9
x = 2.99
x = 2.999
x = 2.9999
f(x)
5
5.9
5.99
5.999
5.9999

x = 4
x = 3.1
x = 3.01
x = 3.001
x = 3.0001
f(x)
7
6.1
6.01
6.001
6.0001


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.