Question

Consider \(\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}+y^{2}}{x y}\). (a) Determine (if possible) the limit along any line of the form \(y=a x .\) (b) Determine (if possible) the limit along the parabola \(y=x^{2}\). (c) Does the limit exist? Explain.

Short answer

The limit along any line of the form \(y = ax\) is \(\frac{1+a^{2}}{a}\) for all a ≠ 0, and the limit along the parabola \(y = x^{2}\) is 0. However, as the limits are different depending on the path taken to approach (0,0), the overall limit of the function does not exist as (x, y) approaches (0,0).

Step-by-step solution

Substitute for y in the function for part (a)

Replace y with ax in the original function: \(\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}+(ax)^{2}}{x(ax)} = \lim _{x \rightarrow 0} \frac{x^{2}+a^{2} x^{2}}{a x^{2}} = \lim _{x \rightarrow 0} \frac{1+a^{2}}{a} =\frac{1+a^{2}}{a}\), valid for all a ≠ 0.

Substitute for y in the function for part (b)

Substitute y with x² in the function: \(\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}+(x^{2})^{2}}{x(x^{2})} = \lim _{x \rightarrow 0} \frac{x^{2}+ x^{4}}{x^{3}} = \lim _{x \rightarrow 0} (x +x) = 0\), when we substitute x = 0 in the equation.

Check for the existence of the overall limit

Use the results from Step 1 and Step 2 to check for the overall limit. Since the function yields different limits along the line \(y = ax\) and the parabola \(y = x^{2}\), we can conclude the limit does not exist as (x, y) approaches (0,0).

Key concepts

Limit Along a Line

In multivariable calculus, calculating the limit of a function as it approaches a point along a specific path is quite insightful. To find the limit along a line, we usually substitute the dependent variable with a term involving the independent variable that represents a line. For instance, if we have a line described by the equation y = ax, where a is the slope, we can substitute y with ax in the function and simplify.

Let's see this in action with the function \( \frac{x^{2}+y^{2}}{xy} \) and examine the limit as \( (x, y) \rightarrow(0,0) \) along this line. After substitution, we arrive at \( \frac{1+a^{2}}{a} \) for any value of \( a \) not equal to zero. This indicates that along any line with a nonzero slope, the limit is a specific finite value that depends on the slope of the line. However, it's important to note that these limits can vary based on the slope, revealing that the overall limit of the function may not exist at that point.

Limit Along a Parabola

Similarly, to find the limit of a function along a parabolic path, we make a substitution in which the dependent variable is set to be a quadratic expression of the independent variable. In our case, considering the parabola \( y = x^2 \), the substitution allows the function to take on a simpler form that evaluates the behavior as it approaches the origin along the curve of the parabola.

After substituting \( y \) with \( x^2 \) in our function, we get \( \lim_{x \rightarrow 0} (x + x) = 0 \) when we take the limit of \( x \) towards zero. This indicates that, along the parabolic path described by \( y = x^2 \), the limit of the function as it approaches the point (0,0) is zero. This special case is interesting as it shows that even though the limit along any line may yield different values, the limit along certain curves might suggest the limit tends towards a specific value, potentially misleading if we don't examine other paths.

Existence of Limit

The existence of a limit in multivariable calculus is a delicate concept. A limit at a point \( (x, y) \) for a function exists only if it approaches the same value regardless of the path taken towards that point. It is not enough for the limit to exist along one line or curve; it must be consistent along all possible paths.

This consistency is crucial. If we find that the value of the limit changes depending on the chosen path, as observed in our exercise where different limits were obtained when approaching (0,0) along a line and a parabola, we can assert that the overall limit at the point does not exist. Establishing the nonexistence of a limit requires that we demonstrate multiple paths yield different limit values, and in such a scenario, there is no single value that the function approaches as it nears the point from all directions.

Multivariable Limits

Multivariable limits are the generalizations of limits in single-variable calculus to functions with several variables. Calculating limits for multivariable functions can be trickier since there can be infinitely many paths along which you can approach a particular point, and analyzing the behavior of the function becomes significantly more complex.

As seen in our example, it is not sufficient to check the limit along just one path; we must consider different paths. Even if a limit appears to exist when approaching from one specific direction, this does not guarantee the limit is the same from another direction. Various techniques, like polar coordinates or the epsilon-delta definition of a limit, might be needed to prove limit existence in the broader, more complex field of multivariable calculus. Ultimately, understanding and determining limits in this context often provides deeper insight into the function's behavior near a point of interest and is crucial for further explorations in calculus.