91Ó°ÊÓ

Understanding the limit of multivariable functions involves grasping how a function behaves as its variables approach certain values. In our case, the function is given by \(f(x, y, z) = x^2 + y^2 + z^2\). To show continuity at the origin, we examine the limit as \((x, y, z)\) approaches \((0, 0, 0)\).

When dealing with multivariable limits, each variable individually approaches its target. Therefore, we watch how \(x\), \(y\), and \(z\) approach 0. A function is well-behaved at a point if the function yields the same result along any path to that point. Here, all paths, including straight lines and curves, lead to \(0\) when approaching \((0, 0, 0)\). Because the squared terms \(x^2, y^2, z^2\) cover any path without negative results, the function appears smooth and unified in its limit.

The Squeeze Theorem is a useful tool for evaluating the limits of functions where direct substitution isn't apt or where comparison can simplify the process. It helps establish the limit by bounding a function between two other functions whose limits are known or easier to find.

For the function \(x^2 + y^2 + z^2\), we know that each squared term is non-negative and small near the origin. Naturally, this suggests:

• \(0 \leq x^2 + y^2 + z^2\)
• alerts us that the function does not dip below 0.

When the inputs \(x, y, z\) are tiny, each squares these small numbers further, driving the entire output toward 0. Thus, by comparing with \(0\), we conclude under the Squeeze Theorem:

• The middle function \(x^2 + y^2 + z^2\) is squashed down to zero.

This helps establish that the limit of \(x^2 + y^2 + z^2\) at the origin is indeed 0. Hence, the theorem effectively seals the entire argument.

Continuity of a function at a point means that the function does not "jump" at this point and is smooth and predictable. Mathematically, for a function \(f(x, y, z)\) to be continuous at \((0, 0, 0)\), the limit of \(f(x, y, z)\) as \((x, y, z) \rightarrow (0, 0, 0)\) should match \(f(0, 0, 0)\).

For the function \(f(x, y, z) = x^2 + y^2 + z^2\), substituting \((0, 0, 0)\) into the function gives \(f(0, 0, 0) = 0\). From previous discussions, we know that as \((x, y, z)\) gets ever closer to \((0, 0, 0)\), \(x^2 + y^2 + z^2\) gets ever smaller, approaching 0.

In essence:

• The proximity of \(f(x, y, z)\) to this value assures continuity.
• Makes sure no sudden changes in the function’s behavior around \((0, 0, 0)\).

Thus, this balance of limits and evaluations assure us of its continuity at the desired point.