Definition: A limit is the value a function approaches as its input gets closer and closer to a chosen number, whether or not the function actually reaches that value there.
What is a limit?
The notation limx→a f(x) = L says that f(x) gets as close to L as you like when x is close enough to a, but not equal to a. The function does not even have to be defined at a.
How do you find one?
Try direct substitution first. If you get 0/0, factor and cancel, or rationalize.
Worked example: limx→3 (x2 – 9)/(x – 3). Substituting gives 0/0. Factor the top: (x – 3)(x + 3)/(x – 3) = x + 3 for x not equal to 3. As x approaches 3, the value approaches 6.
A table agrees: at x = 2.99 the output is 5.99, and at x = 3.01 it is 6.01.
What do students get wrong?
Believing the limit is f(a). The two can differ, or f(a) may not exist. Also, a limit exists only when the left-hand and right-hand limits agree. Mathbot checks both sides before announcing an answer.