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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.
