Limits and Continuity
Limits
Suppose is a function of two variables and is a point in . We say the limit of as tends to is , denoted by
if tends to as tends to i.e. the distance between and tends to .
Intuitively, it means that can be as close to as you want if you choose that is "close enough" to . Actually, the rigorous definition of the limit is as follows:
For any small positive real number , we can find another small positive real number such that whenever the distance from to is smaller than , .
The limit of a function of three variables can be similarly defined - we say that the limit of as tends to is , denoted by
if tends to as tends to .
In the following applet, it shows the idea of limit for a function of two variables:
First fix the point , then set the value of . To check whether , set to be small, and then adjust such that the orange ball centered at point with radius is small enough such that every point in the orange ball satisfies the inequality , which means the corresponding point on the graph is between the two planes and .
(Note: The rigorous definition of limit will not be used in this course.)
Limit along a curve
When we consider the limit of a function of one variable , we can further specify the way approaches . As we know, can approach from the left or right i.e. we have one-sided limits and respectively. As for the limit of a function of two variables , there are many different ways for to approach . This motivates us to consider the following definition:
Suppose a curve is parametrized by on the xy-plane such that . We define the limit of when tends to along the curve C to be
The limit of a function of three variables along a curve can be similarly defined - we say that the limit of as tends to along a curve in is
where is parametrized by with .
Example:
Let for and . Fine the limit of as tends to along each of the following curves:
- is the x-axis i.e.
- is the y-axis i.e.
- is the straight line i.e.
- is the curve i.e.