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- Multivariable Calculus
- Partial Derivatives

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Get Started Now- Intro Lesson: a1:38
- Intro Lesson: b1:42
- Intro Lesson: c5:02
- Intro Lesson: d8:42
- Lesson: 13:25
- Lesson: 23:26
- Lesson: 34:35
- Lesson: 44:57
- Lesson: 56:36

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

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 $(a,b)$:- Plug $(a,b)$ into the function and see if the value is undefined. If it is not undefined, then it is continuous.
- 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.

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.

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

- 1.
**Finding the Limit**

Evaluate the following limit:$\lim\limits_{ (x,y) \to (2,2) } \frac{x-y}{y-x}$

- 2.Evaluate the following limit:
$\lim\limits_{(x,y,z) \to (e,3,4)} \frac{ln(x^2) + \sqrt{y^2 + z^2}}{e^x - 2y + z}$

- 3.Evaluate the following limit:
$\lim\limits_{(x,y) \to (4,1)} \frac{xy^2 - 4y^3}{x^2 - 16y^2}$

- 4.
**Finding Discontinuous Limits**

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

- 5.Evaluate the following limit
$\lim\limits_{(x,y) \to (0,0)} \frac{x^3 - y^6}{xy^2}$