Limits of Functions of Three Variables

We have just looked at Limits of Functions of Two Variables. Recall that for a two variable real-valued function, $z = f(x, y)$, then $\lim_{(x, y) \to (a,b)} f(x, y) = L$ if $\forall \epsilon > 0$ $\exists \delta > 0$ such that if $x \in D(f)$ and $0 < \sqrt{(x - a)^2 + (y - b)^2} < \delta$ then $\mid f(x,y) - L \mid < \epsilon$. We are now going to extend this concept further to functions of three variables.

Definition: Let $w = f(x, y, z)$ be a three variable real-valued function. Then the Limit of $f(x, y, z)$ as $(x, y, z)$ Approaches $(a,b,c)$ is $L$ denoted $\lim_{(x, y, z) \to (a,b,c)} f(x, y, z) = L$ if $\forall \epsilon > 0$ $\exists \delta > 0$ such that if $(x, y, z) \in D(f)$ and $0 < \sqrt{(x - a)^2 + (y - b)^2 + (z - c)^2} < \delta$ then $\mid f(x, y, z) - L \mid < \epsilon$.

Once again, it is important to note that $\sqrt{(x - a)^2 + (y - b)^2 + (z - c)^2}$ represents the distance between the points $(x, y, z)$ and $(a, b, c)$. Thus, we can reformulate the definition above as follows. The limit as $(x, y, z)$ approaches $(a, b, c)$ is the real number $L$ if for all $\epsilon > 0$ there exists a $\delta > 0$ such that if the distance between the points $(x, y, z)$ and $(a, b, c)$ is less than $\delta$ but not $0$, then the distance between $f(x, y, z)$ and $L$ is less than $\epsilon$.

We can also look at limits of functions of more than $3$ variables:

Definition: Let $z = f(x_1, x_2, ..., x_n)$ be an $n$ variable real-valued function. Then the Limit of $f(x_1, x_2, ..., x_n)$ as $(x_1, x_2, ..., x_n)$ Approaches $(a_1, a_2, ..., a_n)$ is $L$ denoted $\lim_{(x_1, x_2, ..., x_n) \to (a_1, a_2, ..., a_n)} f(x_1, x_2, ..., x_n) = L$ if $\forall \epsilon > 0$ $\exists \delta > 0$ such that if $(x_1, x_2, ..., x_n) \in D(f)$ and $0 < \sqrt{(x_1 - a_1)^2 + (x_2 - a_2)^2 + ... + (x_n - a_n)^2} < \delta$ then $\mid f(x_1, x_2, ..., x_n) - L \mid < \epsilon$.