Limits and continuity of multivariable functions

Limits and continuity of multivariable functions



Notation for a Limit of 2 Variable Functions Suppose we want to take the limit of xax \to a and yby \to b of f(x,y). Then we write this notation as:
limit notation

In a sense, we are trying to approach to the point (a,b)(a,b) in the function f(x,y)f(x,y) without being at that point.

Limit of Continuous 2 Variable Functions
A function f(x,y)f(x,y) is continuous at (a,b)(a,b), then:

lim(x,y)(a,b)f(x,y)=f(a,b)\lim\limits_{(x,y) \to (a,b)} f(x,y) = f(a,b)

In other words, if the function is continuous at (a,b)(a,b), then plug in x=ax=a and y=by=b to evaluate the limit!

Ways to see if it is continuous at (a,b)(a,b):
  1. Plug (a,b)(a,b) into the function and see if the value is undefined. If it is not undefined, then it is continuous.
  2. If the function has a denominator, plug (a,b)(a,b) into the denominator and see if it gives 0. If it is not 0, then the function is continuous.

Visualizing the Limit
There are many paths to approaching a specific point. For example, look at this graph:
limit visualization
There are an infinite number of paths into approaching the point (1,0)(1, 0).
One thing to note: If all paths lead to the same value, then the limit is equal to that value.

What if the Function is Discontinuous at (a,b)?
You want to take different paths to (a,b)(a,b) and see if the limits are different. For example, one path can be y=x2y=x^2 and the other path can be y=xy=x. If evaluating these paths with limits give different values, then the limit does not exist! We will show an example of this in the video.
  • Introduction
    Limits & Continuity of Multivariable Functions Overview:
    Notation for a Limit of 2 Variables Functions
    • The limit of f(x,y)f(x,y) as xax \to a, yby \to b
    • What Does This Mean?
    • Approaching the point (a,b)(a,b)

    Visualizing the Limit
    • Can take infinite paths
    • Do all paths give the same value?
    • If it does, then the limit is that value.

    Limit of Continuous 2 Variable Functions
    • If continuous, then plug (a,b)(a,b) into the function
    • How do we know it's continuous?
    • 2 Ways to See Continuity

    What if the Function is Discontinuous?
    • Step 1: See if function is discontinuous
    • Step 2: Find a path
    • Step 3: Find a second path
    • Compare paths

  • 1.
    Finding the Limit
    Evaluate the following limit:

    lim(x,y)(2,2)xyyx \lim\limits_{ (x,y) \to (2,2) } \frac{x-y}{y-x}

  • 2.
    Evaluate the following limit:

    lim(x,y,z)(e,3,4)ln(x2)+y2+z2ex2y+z \lim\limits_{(x,y,z) \to (e,3,4)} \frac{ln(x^2) + \sqrt{y^2 + z^2}}{e^x - 2y + z}

  • 3.
    Evaluate the following limit:

    lim(x,y)(4,1)xy24y3x216y2 \lim\limits_{(x,y) \to (4,1)} \frac{xy^2 - 4y^3}{x^2 - 16y^2}

  • 4.
    Finding Discontinuous Limits
    Evaluate the following limit

    lim(x,y)(0,0)x+yxy \lim\limits_{(x,y) \to (0,0)} \frac{x+y}{x-y}

  • 5.
    Evaluate the following limit

    lim(x,y)(0,0)x3y6xy2\lim\limits_{(x,y) \to (0,0)} \frac{x^3 - y^6}{xy^2}