1. §2.2. Rough definitions of limits
Whether tangent line problems or instantaneous
velocity problems, we are dealing with functions.
So it is necessary to give the definition of a limit
for general functions.
Topics:
Two-sided limits
One-sided limits
Infinite limits
Vertical asymptotes

4. Ex 1:
Ex 2:
Ex 3:

5. In general, we have three approaches to find
the value of a limit:
Numerical approach
Graphical approach
Algebraic approach

6. x
f(x) ?

7. x
½ / ¼ /
f(x)

8. x
/ / / / /21 … 2/(4n+1) ...
f(x)
… …

12. Comparison among these 3 approaches:
Numerical approach: no good, tedious, time-consuming, sometimes, it’s misguiding
So don’t use this approach unless you are asked to use it.
(2)Graphical approach: once you have the graph for the given function, it’s easy to evaluate the limit. But it’s not easy to draw the graph.
So it is not recommended.
(3) Algebraic approach: the most important approach. Most time we use this one.

18. Infinite limits

21. Vertical asymptotes
The line x = a is called a vertical asymptote of y=f(x) if at least one of the following statements is true: