Q 10. - Calculus

Question

Explain how the definition of the limit of a function of two or three variables along a path simplifies to the limit of a function of a single variable 

Step-by-Step Solution

Answer

If the function, giving the relation between $x and y$ is substituted in$f(x,y),$ the function $f$ is reduced to single variable. Thus, the limit of a function of two variables is simplified to single variable limit. 

Step 1: Given Information

Consider the fact that the function f has two variables. Assume $x and y$ are the input variables, and the function is $f(x,y)$. Because the function has two variables, it is in $R^2$. This function's graph will contain three variables. As a result, it will be in $R^3$.

Step 2: Defining the limit

The limit of function is said to be defined at a point. Assume the point to be $(x,y)=(a,b)$ and assume the limiting value to be The limit of function is $\underset{(x,y)\to (a,b)}{\lim }f(x,y)=L$

Step 3: Evaluating the limit

The output that the function tends to approach as the input approaches the point is determined by evaluating the function's limit at a point. A path of approach can be defined when assessing the limit. This path of approach is a function in single variable giving the relation between $x and y$. If this function, giving the relation between $x and y$ is substituted in $f(x,y)$. The function is reduced to single variable. Thus, the limit of a function of two variables is simplified to single variable limit.