Properties of Limits in Multivariable Calculus

In multivariable calculus, the laws of limits provide systematic rules for evaluating limits of functions involving several variables. These laws allow complex expressions to be broken into simpler components whose limits are known. Suppose that

\begin{equation*}\lim_{(x,y)\to(a,b)} f(x,y) = L\end{equation*}

and

\begin{equation*}\lim_{(x,y)\to(a,b)} g(x,y) = M\end{equation*}

where both limits exist, the principal laws are stated as follows.

Sum Law

The limit of a sum equals the sum of the limits:

\begin{equation*}\lim_{(x,y)\to(a,b)} [f(x,y) + g(x,y)] = L + M\end{equation*}

Difference Law

The limit of a difference equals the difference of the limits:

\begin{equation*}\lim_{(x,y)\to(a,b)} [f(x,y) - g(x,y)] = L - M\end{equation*}

Constant Multiple Law

If a function is multiplied by a constant c, the limit is multiplied by the same constant:

\begin{equation*}\lim_{(x,y)\to(a,b)} [c\, f(x,y)] = cL\end{equation*}

Product Law

The limit of a product equals the product of the limits:

\begin{equation*}\lim_{(x,y)\to(a,b)}[f(x,y)g(x,y)] = LM\end{equation*}

Quotient Law

The limit of a quotient equals the quotient of the limits, provided the denominator limit is nonzero:

\begin{equation*}\lim_{(x,y)\to(a,b)} \jfrac{f(x,y)}{g(x,y)} = \jfrac{L}{M}\end{equation*}

where,

\begin{equation*}M \ne 0\end{equation*}

These fundamental properties ensure consistent evaluation of multivariable limits and form the basis for more advanced analytical techniques.