Dated: 30-10-2024

Limit of Multivariable Functions

\[f(x, y) = \sin^{^-1}(x + y)\]

The domain¹ of this function¹ is

\[-1 \le x + y \le 1\]

We can approach a point in space with infinite paths as such

Rules for Non Existence of a Limit

If we have

\[\lim_{(x, y) \to (a, b)} f(x, y)\]

and the function¹ approaches different values as we use different paths then the limit² does not exist.

If we have the following

\[\lim_{(x,y) \to (x_0,y_0)} f(x,y) = L_1\]

\[\lim_{(x,y) \to (x_0,y_0)} g(x,y) = L_2\]

then

\[\lim_{(x,y) \to (x_0,y_0)} cf(x,y) = cL_1; c \in \mathbb{R}\]
\[\lim_{(x,y) \to (x_0,y_0)} (f(x,y) + g(x,y)) = L_1 + L_2\]
\[\lim_{(x,y) \to (x_0,y_0)} (f(x,y) - g(x,y)) = L_1 - L_2\]
\[\lim_{(x,y) \to (x_0,y_0)} (f(x,y) \cdot g(x,y)) = L_1 \cdot L_2\]
\[\lim_{(x,y) \to (x_0,y_0)} \frac{f(x,y)}{g(x,y)} = \frac{L_1}{L_2}\]
\[\lim_{(x,y) \to (x_0,y_0)} c = c; c \in \mathbb{R}\]