The limit of a function of several variables is the value that the function approaches as the input variables approach a specific point. It is the value that the function "approaches" or "tends to" as the input values get closer and closer to the specified point.
The limit of a function of several variables involves multiple input variables, whereas the limit of a single variable function only involves one input variable. Additionally, in a function of several variables, the limit can approach from different directions and still have different values, while in a single variable function the limit will approach from both sides and have the same value.
The limit of a function of several variables is calculated by taking the limit of the function as each individual input variable approaches the specified point. This involves taking multiple limits, one for each input variable, and seeing if they all approach the same value. If they do, then that value is the limit of the function.
The limit of a function of several variables is important in calculus because it allows us to analyze and understand the behavior of a function as the input variables change. This is particularly useful in optimization problems, where we want to find the maximum or minimum values of a function.
Yes, the limit of a function of several variables can exist even if the function is not defined at the specified point. This is because the limit is concerned with the behavior of the function as the input variables approach the specified point, not necessarily the value of the function at that point.