Limits of function of two variables

Limits of Functions of Two Variables

Definition

Let f(x,y) be defined in a neighborhood of the point (a,b), except possibly at the point itself. We say the limit of f(x,y) as (x,y) → (a,b) is L, and write:

lim_{(x,y)→(a,b)} f(x,y) = L

if for every ε>0, there exists δ>0 such that:

0 < √((x-a)² + (y-b)²) < δ ⇒ |f(x,y)-L| < ε

Intuitive Explanation

• The value of f(x,y) gets closer to L as (x,y) approaches (a,b) from any direction.
• The limit must be the same along all paths approaching (a,b).

Examples

Example 1

Function: f(x,y) = x + y

Find: lim_{(x,y)→(1,2)} f(x,y)

Solution: f(1,2) = 1 + 2 = 3

Answer: 3

Example 2

Function: f(x,y) = xy / (x² + y²)

Find: lim_{(x,y)→(0,0)} f(x,y)

Solution:

• Along y=0: f(x,0)=0
• Along x=0: f(0,y)=0
• Along y=x: f(x,x)=1/2

Answer: Limit does not exist (different values along different paths)

Example 3

Function: f(x,y) = x² + y²

Find: lim_{(x,y)→(1,1)} f(x,y)

Solution: Direct substitution: f(1,1) = 1² + 1² = 2

Answer: 2

Example 4

Function: f(x,y) = (x² - y²) / (x² + y²)

Find: lim_{(x,y)→(0,0)} f(x,y)

Solution:

• Along y=0: f(x,0)=1
• Along x=0: f(0,y)=-1

Answer: Limit does not exist

Example 5

Function: f(x,y) = 3x²y / (x² + y²)

Find: lim_{(x,y)→(0,0)} f(x,y)

Solution:

• Along y=0: f(x,0)=0
• Along x=0: f(0,y)=0
• Along y=kx: f(x,kx) → 0 as x→0

Answer: 0

Summary Table

Function	Point	Limit	Exists?
x + y	(1,2)	3	✔
xy / (x² + y²)	(0,0)	–	✘
x² + y²	(1,1)	2	✔
(x² - y²) / (x² + y²)	(0,0)	–	✘
3x²y / (x² + y²)	(0,0)	0	✔

Function of Two variable:

 

Function of Two variable: Let u be a symbol which has a definite value for every pair of values of x and y, then u is called a function of two independent variable x and y and is written as

u=f(x,y)

 

Function of Two Variables – Graphical Representation

A function of two variables is written as:

z = f(x, y)

---

1. General 3D Graph

          z
          |
          |        •
          |      •
          |    •
          |  •
          |•____________ y
         /
        /
       x

Explanation: This shows a surface in 3D space where z depends on x and y.

---

2. Plane Surface (z = x + y)

          z
          |
          |      /
          |    /
          |  /
          |/________ y
         /
        /
       x

Application: Cost, temperature variation.

---

3. Paraboloid (z = x² + y²)

          z
          |
        __|__
      /   |   \
    /     |     \
  /_______|_______\ y
          |
          x

Application: Heat distribution, potential energy.

---

4. Saddle Surface (z = xy)

          z
          |
      ___/ \___
     /           \
----/-------------\---- y
    \             /
      \___   ___/
            x

Application: Profit–loss analysis.

---

5. Contour Diagram (x² + y² = c)

        y
        |
     ○  ○  ○
   ○     ○
 ○    ○    ○
   ○     ○
     ○  ○
        |
        x

Application: Topographic maps, weather maps.

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Exam Note:
The graph of a function of two variables is a surface in three-dimensional space.

Function of Three Variables

A function of three variables is a function that depends on three independent variables.

Mathematical Form:

w = f(x, y, z)

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Examples of Functions of Three Variables

• Linear: f(x, y, z) = x + y + z
• Quadratic: f(x, y, z) = x² + y² + z²
• Product: f(x, y, z) = xyz
• Trigonometric: f(x, y, z) = sin x + cos y + tan z
• Rational: f(x, y, z) = (x + y) / z, z ≠ 0

---

Graphical Representation

A function of three variables cannot be drawn directly because it requires four dimensions. Hence, it is represented using level surfaces or cross-sections.

Level Surface Example

For: x² + y² + z² = c (Sphere)

          z
          |
       ___|___
    .-'    |    '-.
  .'        |        '.
 |          |          |
  '.        |        .'
    '-._____|_____.-'
          |
          x
         /
        y

Explanation: Each surface represents points where the function has the same value.

---

Cross-Section Concept

Fix one variable (e.g., z = k) and draw the 3D surface in x–y plane.

        z = k
          |
      ____|____
     /    |    \
    |     |     |
     \____|____/
          |
          x
         /
        y

---

Real-Life Applications

• Temperature in a room: T = f(x, y, z)
• Pressure in fluids
• Electric and gravitational potential
• Air pollution concentration
• Medical imaging (CT, MRI)

---

Exam Note:
A function of three variables is represented graphically using level surfaces or cross-sections.

Assignment

Functions of Two and Three Variables

Section 1: Functions of Two Variables

Definition

A function of two variables depends on two independent variables x and y and produces a single dependent variable z:

z = f(x, y)

Examples

• f(x, y) = x + y
• f(x, y) = x² + y²
• f(x, y) = xy
• f(x, y) = sin x + cos y
• f(x, y) = x / y, y ≠ 0

Graphical Representation

• Graph is a surface in 3D space.
• Plane: z = x + y
• Paraboloid: z = x² + y²
• Saddle: z = xy

Level Curves

A level curve is given by f(x,y) = c, e.g., x² + y² = 1.

Applications

• Temperature distribution
• Population density
• Cost analysis
• Pressure on surfaces
• Topography

Section 2: Functions of Three Variables

Definition

A function of three variables depends on x, y, z and produces a single output w:

w = f(x, y, z)

Examples

• f(x,y,z) = x + y + z
• f(x,y,z) = xyz
• f(x,y,z) = x² + y² + z²
• f(x,y,z) = sin x + cos y + tan z
• f(x,y,z) = (x+y)/z, z ≠ 0

Graphical Representation

• Cannot be plotted directly in 3D (requires 4D).
• Use level surfaces: f(x,y,z) = c
• Use cross-sections: fix one variable and plot remaining 2D function

Applications

• Temperature in a room
• Pressure in fluids
• Air pollution modeling
• Electric/gravitational fields
• Medical imaging (CT/MRI)

Section 3: Questions and Answers

Functions of Two Variables

Q1: Define a function of two variables.

A1: A function of two variables depends on x and y and produces a single output z.

Q2: Write the general form of a function of two variables.

A2: z = f(x, y)

Q3: How many dimensions are needed to graph it?

A3: Three dimensions (x, y, z)

Q4: What is a level curve? Give an example.

A4: A curve where f(x,y) = c, e.g., x² + y² = 1

Q5: Give any two examples.

A5: f(x,y)=x+y, f(x,y)=x²+y²

Q6: If f(x,y)=x²+y², find f(2,3).

A6: f(2,3)=2²+3²=13

Q7: Find f(-2,4) for f(x,y)=xy.

A7: f(-2,4)=(-2)(4)=-8

Q8: Determine domain of f(x,y)=x/y.

A8: All real x,y with y≠0

Functions of Three Variables

Q1: Define a function of three variables.

A1: Depends on x, y, z and produces a single output w.

Q2: Write the general form.

A2: w = f(x, y, z)

Q3: Why cannot it be plotted directly?

A3: Requires 4D visualization; use level surfaces or cross-sections.

Q4: Give three examples.

A4: f(x,y,z) = x+y+z, f(x,y,z) = xyz, f(x,y,z)=x²+y²+z²

Q5: What is a level surface? Give example.

A5: Surface where f(x,y,z)=c, e.g., x²+y²+z²=1 (sphere)

Q6: If f(x,y,z)=x+y+z, find f(1,2,3).

A6: 6

Q7: If f(x,y,z)=xyz, find f(2,-1,3).

A7: -6

Q8: Determine domain of f(x,y,z)=(x+y)/z.

A8: All real x,y,z with z≠0

Section 4: Diagrams

General Graph of Function of Two Variables z | | • | • | • | • |•____________ y / / x

Plane Surface (z = x + y) z | | / | / | / |/________ y / / x

Paraboloid (z = x² + y²) z | __|__ / | \ / | \ /_______|_______\ y | x

Saddle Surface (z = xy) z | ___/ \___ / \ /-------------\---- y \ / \___ ___/ x

Level Curve (x² + y² = c) y | ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ | x

Level Surface for Function of Three Variables (x² + y² + z² = c) z | ___|___ .-' | '-. .' | '. | | | '. | .' '-.__|__. -' | x / y

Cross-Section for Function of Three Variables z = k | ____|____ / | \ | | | \____|____/ | x / y

Calculus of Several Variables

 Calculus of Several Variables

Table of Content 

1	Functions of two variables
2	Limits of function of two variables
3	Continuity of function of two variables
4	Partial derivatives
5	Partial derivatives
6	Partial derivatives
7	Total derivatives
8	Total derivatives
9	Maxima and minima
.          10	Lagrange multipliers method.