91Ó°ÊÓ

A limit is a value that a function approaches as the input approaches a certain point. Mathematically, the limit of a function f(x,y) as (x,y) approaches (a,b) is expressed as: \[\lim_{(x, y)\to (a, b)}f(x, y) = L\] If the value of the function approaches L no matter which route is taken to approach (a,b), we say that the limit exists.

Multiple paths are different routes along which the input values approach the desired point in the domain. For instance, given a function of two variables, various paths can be taken to approach a specific point (a, b), such as along a line, a curve, or a circle. Differentiating along various paths will help assess the function's behavior and limit existence.

Let's consider the following function as our example: \[f(x, y) = \frac{xy}{x^2 + y^2}\] We want to evaluate the limit of the function as (x, y) approaches (0, 0): \[\lim_{(x,y) \to (0,0)} \frac{xy}{x^2 + y^2}\]

Let's first examine the limit along the straight line path y = mx, where m is a constant: \[f(x, mx) = \frac{mx^2}{x^2 + m^2x^2} = \frac{mx}{1 + m^2}\] Now, let's find the limit as x approaches 0: \[\lim_{x\to 0} \frac{mx}{1 + m^2} = 0\] The above limit suggests that the limit exists and equals 0 along all the straight line paths y = mx. However, let's consider another path: the curve y = x^2. The function becomes: \[f(x, x^2) = \frac{x^3}{x^2 + x^4} = \frac{x}{1 + x^2}\] Now, let's find the limit as x approaches 0: \[\lim_{x\to 0} \frac{x}{1 + x^2} = 0\] This limit also equals 0, which could make us believe that the limit of the function exists. However, we forgot an important path: along the x-axis (y=0) and y-axis (x=0). Let's examine these paths as well.

When y=0, the function becomes: \[f(x, 0) = \frac{0}{x^2} = 0\] And when x=0, the function becomes: \[f(0, y) = \frac{0}{y^2} = 0\]

After examining various paths to approach (0, 0), we find that the limit is always equal to 0. In this case, the limit existence has been proven by examining multiple paths. However, if we had found a path where the limit was unequal, or it did not exist, that path would have disproved the existence of the limit simply because the function does not approach the same value along all paths. So, to answer the question "how examining limits along multiple paths may prove the nonexistence of a limit," the answer is: by showing that the function does not approach the same value or no value along particular paths. If different paths lead to various values or no value, it indicates that the limit does not exist at that point.