Question

In Exercises \(33-38,\) use a computer algebra system to graph the function and find \(\lim _{(x, y) \rightarrow(0,0)} f(x, y)\) (if it exists). \(f(x, y)=\frac{x^{2} y}{x^{4}+4 y^{2}}\)

Short answer

After finding the limit of the given function as (x, y) approaches (0,0), it's essential to cross verify it with the plot obtained from a computer algebra system to ensure we have correctly determined the limit.

Step-by-step solution

Substitute the limit point into the function

First it's necessary to substitute x = 0 and y = 0 into the function \(f(x, y)=\frac{x^{2} y}{x^{4}+4 y^{2}}\). Simplify the expression after substituting the variables.

Determine the limit

If the expression after substitution in Step 1 does not yield an indeterminate form, then that value is the limit. If it yields an indeterminate form, such as 0/0 or infinity/infinity, further analysis would be necessary, either by simplifying the function or by approaching the limit from different directions and checking if the limit exists.

Visualize the function with a graph

Plot the graph of the function using a computer algebra system to visualize its behavior near the point (0,0). The graph will help us better understand how the function behaves around this point and ascertain whether or not the limit exists.