Before introducing Differentiation, it would be useful to first understand the concept of Limits.

Visit http://www.calculus-help.com/tutorials, and watch the flash tutorials on Chapter 1: Limits & Continuity. You only need to view Lessons 1 to 5.


Limits_Definition.jpg

Here are 2 different ways of determining the limit of a particular function:


(1) Using a Table


Study the table. What happens to the values of a function f(x) as x approaches 1

x
0.9
0.9999
0.99999
0.999999
f(x)
2
2.899
2.9999
2.99999

(2) Using a Graph


Similarly, study the following graphs and determine the limit of the functions:
Limit_from_graph.jpg


The difference between “approaching a value” and “equals a value”:

diff_bet_equal_and_app.jpg


When do Limits exist?


For any given function, a limit exists if and only if the same value is approached from both the left and right side. [Note that it does not matter even if the value does not exist!]

In other words,
exist1a.jpg
Conversely,
exist1.jpg


How to find the limit?


Example 1: Using a table of values (tedious!)



Example 2: Using a graph (troublesome!)

limit_example_2.jpg


Apart from using tables and graphs, there are 3 other ways to evaluate the limits of functions.
(1) By Substitution
Example:
sub_example.jpg
Always try substitution first!

(2) By Factorisation
Example:
diff_bet_equal_and_app.jpg
If substitution leads to an undefined result, try factoring to see if you are able to simplify the function.

(3) By multiplying a conjugate
conjugate_example.jpg
This method is more useful when dealing with radical/surd form.


Limits at Infinity

limit_infinity.jpg




Limits and Asymptotes:

vertical_asymptote.jpg

horizontal_asymptote.jpg

In other words,


summary.jpg