Q. 16 - Calculus

Question

Provide a definition for $\underset{(x,y,z)\to (a,b,c)}{\lim }f(x,y,z)=\infty$. Model your definition on Definitions 1.9 and 12.15. 

Step-by-Step Solution

Answer

For all $M>0$, there exist $\delta >0$ such that if $0<\sqrt{{(x-a)}^2+{(y-b)}^2+{(z-c)}^2}<\delta$, then $f\left(x\right)\in (M,\infty )$

Step 1: Defining the limit

The goal is to provide a definition for $\underset{(x,y,z)\to (a,b,c)}{\lim }f(x,y,z)=\infty$.

The infinite limit at a point, represented as $\underset{x\to a}{\lim } f\left(x\right)=\infty$, states that for any $M>0$, there exists $\delta >0$ such that $x\in (a-\delta ,a)\cup (a,a+\delta )$, then $f\left(x\right)\in (M,\infty )$ . L is the limit of a function with two or more variables, represented as $\underset{x\to a}{\lim }f\left(x\right)=L$ ,if there is $\epsilon >0$ then there is $\delta >0$ such that $\left|f\left(x\right)-L\right|<\epsilon$ whenever$0<\left|\left|(x-a)\right|\right|<\delta$.

Step 2: Evaluating the limit

To create the definition of infinite limit of a function with two variables, combine the two definitions.

The infinite limit of a function f in three variables at a point, expressed as $\underset{(x,y,z)\to (a,b,c)}{\lim }f(x,y,z)=\infty$, indicates that there exists $\delta >0$ such that if  $0<\sqrt{{(x-a)}^2+{(y-b)}^2+{(z-c)}^2} <\delta$then $f\left(x\right)\in (M,\infty )$ for every $M>0$.