Calculus
Lesson
03
Introduction to Limits
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DON'T
PANIC!!!
Our goal is to find HOW A FUNCTION BEHAVES NEAR some point.
For instance:
What value does 
approach
as 
gets close to zero from
either side?
Since 
, we say 
What value does 
approach
as 
gets close to three
from either side?
Since 
, we say 
What value does 
approach
as 
gets close to negative
2 from either side?
Since 
, we say 
Note, however, that there is no value for 
, since -2 is not in the
domain! However, all we care about for limits, is what does
the value of the function approach as x approaches some number.
In evaluating the limit, we never actually reach -2, so we never
divide by zero! We just get really close to -2! When
we do so, f gets really close to 12. That's all we're
saying!!!
We can also look at limits graphically:
 |
The graph approaches the same value from either side, so we know what the limit is!
 |
The graph does NOT approach the same value from either side, so there is NO LIMIT!
Lastly, we could evaluate the limit analytically, which we will learn how to do during the next lesson. First, let's look at some special cases:
1) Point Discontinuity: there is a gap only at one point. There is actually a limit in this case.
We can read it straight from the graph if we've been given the numbers.
 |
2) Limit does not exist: 
From the right: 
From the left: 
What is the limit? NO LIMIT
And graph it....
3) Unbounded: 
: NO LIMIT
4) Oscillating: 
look
at graph on calculator.