3.3 Limits

Limits

Introduction

Limits of Functions of Two Variables

In this Preview Activity, we will recall and work with several ideas related to limits of single-variable functions by working with tables, graphs, and the algebraic forms of these functions.

Let $f$ defined by $f(x) = 3-x$. Complete the table below.

$x$ | $-0.2$ | $-0.1$ | $-0.01$ | $0.01$ | $0.1$ | $0.2$

$f(x)$ | | | | | |

Work through this exercise and explain your reasoning step by step.

Is $f$ defined at the point $(0,0)$? What, if anything, does this say about whether $f$ has a limit at the point $(0,0)$?

Values of $f$ (to three decimal places) at several points close to $(0,0)$ are shown below.

$x\backslash y$ | $-1.000$ | $-0.100$ | $0.000$ | $0.100$ | $1.000$

$-1.000$ | $-0.707$ | —— | $0.000$ | —— | $0.707$

$-0.100$ | —— | $-0.707$ | $0.000$ | $0.707$ | ——

$0.000$ | $-1.000$ | $-1.000$ | —— | $1.000$ | $1.000$

$0.100$ | —— | $-0.707$ | $0.000$ | $0.707$ | ——

$1.000$ | $-0.707$ | —— | $0.000$ | —— | $0.707$

Based on these calculations, state whether you think $f$ has a limit at $(0,0)$ and give an argument supporting your statement.

Now we formalize your conjecture from the previous part by considering what happens if we restrict our attention to different paths. First, we look at $f$ for points in the domain along the $x$-axis. That is, we consider what happens when $y = 0$. Write out what the output of $f$ will when $y=0$. In other words, what is $f(x,0)$ equal to?

What is the behavior of $f(x,0)$ as $x \to 0$? If we approach $(0,0)$ by moving along the $x$-axis, what value do we find as the limit?

What is the behavior of $f$ along the line $y=x$ when $x \gt 0$. That is, what is the value of $f(x,x)$ when $x>0$? If we approach $(0,0)$ by moving along the line $y=x$ in the first quadrant, what value do we find as the limit?

In general, if $\displaystyle{\lim_{(x,y)\to(0,0)}f(x,y) = L}$, then $f(x,y)$ approaches $L$ as $(x,y)$ approaches $(0,0)$, regardless of the path we take in letting $(x,y) \to (0,0)$. Based on the last two parts of this activity, can the limit

exist? Write a sentence or two to explain your reasoning.

Substitute $y=mx$ into $f(x,y)$ and simplify your result for $f(x,mx)$. Use your expression and limit rules to evaluate

. Use this limit to explain the the behavior of $f(x,y)$ as $(x,y)$ approaches $(0,0)$ along lines of the form $y=mx$ and how this observation reinforces your conclusion about the existence of $\displaystyle{\lim_{(x,y)\to(0,0)}f(x,y)}$ from the previous part of this activity.

What is the behavior of $g$ on the $x$-axis? That is, what is $g(x,0)$ and what is the limit of $g$ as $(x,y)$ approaches $(0,0)$ along the $x$-axis?

What is the behavior of $g$ on the $y$-axis? That is, what is $g(0,y)$ and what is the limit of $g$ as $(x,y)$ approaches $(0,0)$ along the $y$-axis?

What is the behavior of $g$ on the line $y=mx$? That is, what is $g(x,mx)$ and what is the limit of $g$ as $(x,y)$ approaches $(0,0)$ along the line $y=mx$?

Based on what you have seen so far, do you think $\displaystyle\lim_{(x,y)\to(0,0)}g(x,y)$ exists? If so, what do you think its value is?

Now consider the behavior of $g$ on the parabola $y=x^2$? What is $g(x,x^2)$ and what is the limit of $g$ as $(x,y)$ approaches $(0,0)$ along this parabola?

State whether you think the limit $\displaystyle{\lim_{(x,y)\to(0,0)} g(x,y)}$ exists or not and provide a justification of your statement.

Work through this exercise and explain your reasoning step by step.