Limit of a function of two variables

 Goal: define the limit of a function of two variables

The limit of a function of two variables is based on the limit of a function of one variable. Let's recall the definition:

Let f(x) be defined for all x≠a in an open interval containing a . Let L be a real number. Then 

if  for every ϵ> 0 there exists ẟ> 0 such that if 0< ❙x - a❙ < δ for all x in the domain of f, then

The idea of an open interval in a function of two variables is similar to the open interval in a single variable. Let's define an open interval in a function of two variables.

Definition 
An open disk δ centered at a point (a, b) ε R² and of radius  δ is defined by:

In a function of one variable, when x is is very close to a, we have ❘x-a❘ < δ meaning that the distance of all x to a is very small. In a disk, the distance of all circles are smaller than the square of the radius δ

If we take the square of both sides and taking into account that the first side is strictly positive, we have:

Definition

Let f be a function of two variables x and y. The limit of the function f(x, y) as (x, y) approaches (a, b)) is L  written as:

if for every ϵ>0, there exists a small enough δ>0 such that for all points (x, y) in a delta disk around (a, b) except for (a, b) itself, the value of f(x,y) is no more away than ε from L. Using symbols, we rite:

for every ϵ > 0, there exists ẟ > 0 such that

 Theorem

Limit rules

Let f(x,y) and g(x,y) be defined for all (x, y) ≠ (a, b) in a neighborhood around (a, b), and assume the neighborhood is completely inside the domain of f. Assume L and M are real numbers such that

,

Then each of the following statements is true:

Constant rule

Identity rules

Sum rule

Difference rule

Constant multiple rule

Product rule

Quotient rule

Power rule

for any positive integer n                                                                                                                               

Root rule:

for all L if n is odd and positive and for all L⪈ 0 if n is even and positive for L≽ 0 for all (x,y) ≠ (a, b)

in neighborhood of (a, b)

Example 

Find each of the following limits: 

Solution

Let's first use the sum and difference rules to separate the terms:

                                                                                                                     

Let's use the constant multiple rule on the second, third, fourth and fifth limit:

Let's use the power rule on the second and third limit then the product rule on the second limit

Let's use the identity rule on the first six limits and the the constant rule on the last limit:

b. In this example we need to make sure that the limit of the denominator is different of zero. Upon calculating the limit of the denominator we find:

It is different of zero. Now let's calculate the limit of the numerator:

Finally applying the quotient rule, we find:

Practice

Evaluate the following limit: