Published On:Wednesday, 14 December 2011
Posted by Muhammad Atif Saeed
Limit and Limit Laws
The definition is difficult to use in practice, thus there exist properties that allow us to calculate the limits
- A limit of a sum is the sum of the limits
- A limit of a product is the product of the limits
- The limit of a quotient is the quotient of the limits
- The limit of a constant
is a constant
- The limit of a constant coefficient
of a function 
is the limit of the function multiplied by the coefficient
- If is continuous at
and 
then
- We can derive other limits.
Given
where 
is constant and belongs to the set of rational numbers 
, then
- We can derive other limits.
- If
for all 
then if 
and 
approach some limit 
, 
, then must approach the same limit 
- Left and Right Limits
- Another way to address a limit at a point is to address the limit defined by the behavior of the function to the left of the point
and to the right of the point 
- CONSIDER: The limits of the heaviside function
at 
The limit defined by the behavior of
to the left of is given by
The limit defined by the behavior of to the right of is given by
The limit defined by the overall behavior of at is given by the original definition of the limit. As such it does not exist
- CONSIDER: The limits of the heaviside function
- Left and right limits allow us to predict the behavior of discontinuous functions, or functions that are continuous on their domain, but contain asymptotes
Limits at Infinity- Limits at infinity for a continuous defined on
, is similar to the limits of a sequence as it approaches infinity
- Similarly the limit at negative infinity (
) for a continuous function defined on is simply the reversal of the inequality
"There exists a value
such that whenever 
then 
"
- A graphical representation is the graph of
We know that as
then 
We can write
in the form 
thus, 
The formal epsilon definition simply states for an arbitrarily chosen
, there exists a value , such that whenever is greater than , then the value of are always less than the range of values created from
- The value of is dependant on our choice of
- Continuity
- A function if the following holds
- CONSIDER: The Heaviside function , is it possible for any chosen to force two values
and 
to be close such that 
?
If and are the same sign (the same sign property 
) then no matter how close we choose and , they will always satisfy the condition
If and are opposite signs 
then the condition is dependant on our choice of . If epsilon is chosen such that 
then the condition holds regardless of how we choose and
However, if
then the condition does not hold if and are opposite signs.
- The definition of a continuous function is essentially a limit
- A function is continuous on an interval if it is continuous at each point on the interval
DEFINITIONFor inputs
and 
in order for 
to be so close to 
so that 
(for an arbitrarily chosen 
), it suffices that and are soo close that 
(where 
is dependant on the choice of ) DEFINITIONA function is continuous at
if
Implying that
- A function
