Question

Use a computer to graph the function\(f(x,y) = \frac{{x + y}}{{{x^2} + {y^2}}}\)using various domains and viewpoints. Comment on the limiting behavior of the function.What happens as both x and y become large? What happens as\((x,y)\)approaches the origin?

Short answer

The values of function \(f(x,y) = \frac{{x + y}}{{{x^2} + {y^2}}}\) approach \(0\)as \(x,y\) become large; as \((x,y)\) approaches the origin, function value approaches \( \pm \infty \)or \(0\) depending on the direction of approach.

Step-by-step solution

Analyze the graph

\(f(x,y) = \frac{{x + y}}{{{x^2} + {y^2}}}\)

Let us first look to see what happens near the origin

Clearly, as \((x,y) \to (0,0)\) , the graph exhibits asymptotic behavior. For \(x + y > 0\) the function values move towards \(\infty \) and for \(x + y < 0\), the function values move towards \( - \infty \).

Look at the end behavior of the graph

When x and y both get large, the function values approach \(0\)as \(x,y\)become large; as \((x,y)\)approaches the origin, function approaches \( \pm \infty \)or \(0\) depending on the direction of approach.

Hence, The values of function \(f(x,y) = \frac{{x + y}}{{{x^2} + {y^2}}}\) approach \(0\) as \(x,y\) become large; as \((x,y)\) approaches the origin, function value approaches \( \pm \infty \)or \(0\) depending on the direction of approach.