From 7507725c715397062e8804921cbfed3a5142db0a Mon Sep 17 00:00:00 2001 From: eggy Date: Wed, 11 Jan 2023 15:39:46 -0500 Subject: [PATCH] math119: add limits --- docs/1b/math119.md | 30 ++++++++++++++++++++++++++++++ 1 file changed, 30 insertions(+) diff --git a/docs/1b/math119.md b/docs/1b/math119.md index 4a959d8..a49366b 100644 --- a/docs/1b/math119.md +++ b/docs/1b/math119.md @@ -34,3 +34,33 @@ In order to create a sketch for a multivariable function, this site does not hav - Sketch a two-dimensional graph with constant values of $k$ (e.g., $k=-2, -1, 0, 1, 2$) using the other two variables as axes Combine the three **contour plots** in a three-dimensional plane to form the full sketch. + +A **hyperbola** is formed when the difference between two points is constant. Where $r$ is the x-intercept: + +$$x^2-y^2=r^2$$ + + + +If $r^2$ is negative, the hyperbola is is bounded by functions of $x$, instead. + +## Limits of two-variable functions + +A function is continuous at $(x, y)$ if and only if all possible lines through $(x, y)$ have the same limit. Or, where $L$ is a constant: + +$$\text{continuous}\iff \lim_{(x, y)\to(x_0, y_0)}f(x, y) = L$$ + +In practice, this means that if any two paths result in different limits, the limit is undefined. Substituting $x|y=0$ or $y=mx$ or $x=my$ are common solutions. + +!!! example + For the function $\lim_{(x, y)\to (0,0)}\frac{x^2}{x^2+y^2}$: + + Along $y=0$: + + $$\lim_{(x,0)\to(0, 0) ... = 1$$ + + Along $x=0$: + + $$\lim_{(0, y)\to(0, 0)} ... = 0$$ + + Therefore the limit does not exist. +