Limits and continuity of multivariable functions

Lessons

Notes:

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

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

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

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

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

Ways to see if it is continuous at $(a,b)$:

1. Plug $(a,b)$ 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 $(a,b)$ 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:

There are an infinite number of paths into approaching the point $(1, 0)$.
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 $(a,b)$ and see if the limits are different. For example, one path can be $y=x^2$ and the other path can be $y=x$. 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:
  a)

  Notation for a Limit of 2 Variables Functions

  • The limit of $f(x,y)$ as $x \to a$, $y \to b$
  • What Does This Mean?
  • Approaching the point $(a,b)$
  
  
  b)

  Visualizing the Limit

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

  Limit of Continuous 2 Variable Functions

  • If continuous, then plug $(a,b)$ into the function
  • How do we know it's continuous?
  • 2 Ways to See Continuity
  
  
  d)

  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
  
  

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• 1.
  Finding the Limit
  Evaluate the following limit:

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

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• 2.
  Evaluate the following limit:

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

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• 3.
  Evaluate the following limit:

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

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• 4.
  Finding Discontinuous Limits
  Evaluate the following limit

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

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• 5.
  Evaluate the following limit

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

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