     Calculus Animations,Graphics and Lecture Notes Visualizing Limits of Functions of 2 Variables    Home About Kelly Liakos Sponsors Calculus 1- Limits and Derivatives Calculus 1 - The Second Derivative Calculus 1 and 3 Formula Sheets Chain Rule Computer Lab Assignments Conic Sections Differential Equations Directional Derivatives/Gradient Double Integrals Equillibrium Solutions 1st order DEs 1st order Diff Eqs -Motion Flux Integrals and Surface Integrals Infinite Sequences Infinie Series Level Curves and Level Surfaces Line Integrals Optimization and Related Rates Optimization for Functions of 2 Variables Parametric Equations 2-space Parametric Equations 3-space Partial Derivatives Polar Coordinate System Polar Coordinates- Derivatives and Integrals PreCalculus Riemann Sums and the Fundamental Theorem of Calculus 2d order Diff EQS-Motion 2d Partial Derivatives Supplemental Exercises and Solutions Tangent Planes/ Differential for f(x,y) Trigonometry Trigonometry Applications Triple Integrals Unit Tangent Vectors/Unit Normal Vectors Vectors in General Vector Valued Functions Visualizing Limits of Functions of 2 Variables Work Links Special Topics Scratch Paper 2 Poems   The basic idea for limits of functions of one variable is that given any ε we can find a δsuch that if  we are within  δ of x = a in the domain then f(x) lies between the lines y = L - ε and y = L + ε  . Formally  | f(x) - L| < ε  whenever | x-a | <  δ.But what about a function of 2 variables?  I'll formalize the language later but for now we'll consider the following 3 animationswhich illustrate qualitatively what limits of functions of 2 variables are all about.The first difference when we talk about functions of 2 variables is that instead of considering an interval containing x = a in the domainof the form a - δ < x < a + δWe talk about a circle centered at a point (x0,y0) of radius δ  in the domain of the form (x-x0)2 + (y-y0)2 < δ2 .Then the definition of limit takes on a strikingly similar form: The limit of f(x,y) as (x,y) approaches (x0,y0)  = L  means thatGiven any ε there is a  δ such that  the value of z = f(x,y)  lies between the planes z = L - ε and z = L + εWhenever (x,y) is in the circle  (x-x0)2 + (y-y0)2 < δ2 .  See Animation 1 -- Notice as we shrink the circle (let δ go to 0) centered at (x0,y0) the surface is confined between planes which are converging to z = 1. Animation 1 Another way of thinking of limit is that if we approach (x0,y0) along any curve in thedomain we approach the same value on the surface. In the following animation we consider thesurface z = 1 - x2 - y2  (a parabaloid) , We consider what happens as we approach (0,0) along four different lines. Note on the surface all values converge to 1. See Animation 2 Animation 2 Of course this is not a proof --we can't possibly consider the limit of every curve which approaches a point ( x0,y0). The formal  proof goes like this: We want to show z =1- x2 -y2 goes to 1 as (x,y) goes to (0,0)Let ε be given then | f(x,y) - L | = | (1 - x2 - y2) -1| = |x2 + y2|Let δ =√ ε    then if  x2 +  y2  <  δ2             | f(x,y) - L | < ε     We can though use this to prove a limit does not exist as the next animation shows. If on the surface z approaches different values as we approach (x0,y0) along different paths then we say the limit does not exist.See animation 3. You'll notice that all 3 lines approach (0,0) in the domain but on the surface the corresponding z values approach 3 different values.                                                , Animation 3 Why doesn't this limit  exist ?Here f(x,y) = (xy)/(x2 + y2)suppose we approach (0,0) along any line of the form y = m xf(x,mx) = mx2/[x2(m2+1)] = m/(m2+1)  hence the limit is m/(m2+1)  which can take on any value in  [-1/2,1/2]  The Smart Bunny-A very short story by Kurt Vonnegut Jr.  