Limit and Continuity

LIMIT AND CONTINUITY

Definition 3.1.1 Function of two variables Let S be subset of R2. A function f: S→ R is a rule which assigns to every (x, y) ∈ S a unique real number in R, denoted by f(x, y).

We say f(x, y) is a function of two independent variables x and y.

S is called the domain of the function f.

Example 1 If f(x, y) = find the domain and ƒ(1, 3).

Solution 

Domain of ƒ is the set of all points in the plane at which f(x, y) exists. f(x, y) is defined for all x ≠ y

So domain D = {(x, y) ∈ R2 | x ≠ y}

Geometrically, D is the xy-plane, except the line y = x.

Neighbourhood of a point in the plane

Definition 3.1.2 The δ -neighbourhood of the point (a, b) is the disc

A neighbourhood may also be taken as a square 0 < |x − a| < δ, 0 < | y − b| < δ

Limit of a function

Definition 3.1.3 Let f be a function defined on S ⊂ R2. The function f is said to tend to the limit l as (x, y) → (a, b) if to every ɛ > 0, ∃ δ > 0, such that |f(x, y) − l < ɛ, for all (x, y) satisfying ||(x, y) − (a, b)|| < δ

This limit is called the double limit or simultaneous limit of f(x, y)

Note

(1) If for every (x, y) ∈ S ⊂ R2, there is a unique z assigned by f, then z = f(x, y). Geometrically this represents a surface.

(2) If f(x, y) = l and if y = φ (x) is a function such that φ (x)→ b 

(3) To test limit f(x, y) does not exist.

If l1 ≠ l2, then the limit of the function does not exist.

Example 2 

Solution 

This depends on m and so for different values of m, we will get different limit values.

Hence the limits along different paths are different.

⸫ the limit does not exist.

Note The existence of does not imply the existence of

Repeated limits or iterated limits

Definition 3.1.4 If f(x, y) is defined in a neighbourhood of (a, b) and if  exists, then the limit is a function of y and the limit as y → b is written as f(x, y). This limit is called repeated limit of f(x, y) as x → a a first and then as y → b.

Similarly, we can definite the repeated limit  f(x, y). The two repeated limits may or may not exist and when they exist, they may or may not be equal. Even if the repeated limits have the same value, the double limit may not exsist.

Remark If the double limit f(x, y) exists, then we cannot say repeated limits exist. But if the repeated limits exist and are not equal, then the double limit cannot exist.

(2) If the double limit exist and repeated limits exist, then they are equal. 

That is

Example 3 If f(x, y) = where (x, y) ≠  (0, 0), find the repeated limits and double limit, if they exist

Solution 

Since the repeated limits are unequal, double limit does not exist.

Continuity of a function

Definition 3.1.5 A function f(x, y) is said to be continuous at (a, b) if f(x, y) = f(a, b)

WORKED EXAMPLES

Example 1 Text the continuity of the function

at the origin.

Solution 

Hence f(x, y) is continuous at (0, 0).

Example 2

Solution

The limit depends on m and so for different values of m, we get different limits. Hence the limit does not exist.

Example 3 Find the limit and test for continuity of the function.

Solution 

By the definition of the function ƒ(0, 0) = 0

Since the limit depends on m, for different values of m, we will have different

limit values. Hence limit does not exist.

⸫ The function is not continuous at (0, 0).

EXERCISE 3.1

1. Evaluate the following limits, if they exist.

2. Test continuity of the following

ANSWERS TO EXERCISE 3.1

1. (i) does not exist. (ii) does not exist (iii) does not exist.

2. (i) not continuous

(ii) not continuous

[Hint limit does not exist. Choose paths y = x, x = y3] 

(iii) not continuous