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∞calculus iv review

F(x,y) = sin(xy)

Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025

Definition

The function f(x,y) = sin(xy) is a two-variable function that takes inputs x and y and produces an output based on the sine of the product of x and y. Understanding this function is crucial for analyzing limits and continuity in multiple variables, as it provides insight into how the function behaves as the inputs approach specific points in the domain.

5 Must Know Facts For Your Next Test

  1. The sine function oscillates between -1 and 1, so the output of f(x,y) = sin(xy) will also always be within this range for any real values of x and y.
  2. To analyze limits of f(x,y) as (x,y) approaches a point (a,b), you can use various paths to see if the limit is consistent regardless of the approach.
  3. If either x or y approaches zero, then f(x,y) approaches sin(0), which equals 0, showing that (0,0) could be a limit point.
  4. Continuity at a point (a,b) means that the limit of f(x,y) as (x,y) approaches (a,b) must equal f(a,b).
  5. The multivariable function f(x,y) = sin(xy) is continuous everywhere in its domain, as it is composed of continuous functions.

Review Questions

  • How would you determine if f(x,y) = sin(xy) is continuous at the point (0,0)?
    • To determine if f(x,y) = sin(xy) is continuous at (0,0), first compute f(0,0), which equals sin(0) = 0. Next, evaluate the limit of f(x,y) as (x,y) approaches (0,0). You can approach (0,0) along different paths, like setting y = kx or x = ky, and find that the limit consistently yields 0. Since both the limit and the value of the function at that point match, f(x,y) is continuous at (0,0).
  • Explain how the behavior of f(x,y) = sin(xy) changes when one of the variables approaches zero.
    • When either variable x or y approaches zero, the product xy also approaches zero. This means that f(x,y) = sin(xy) approaches sin(0), which equals 0. As you analyze different scenarios where either x or y gets very close to zero while keeping the other variable fixed, you'll see that no matter how you approach it, the output consistently trends towards 0. This showcases how small input changes lead to predictable outputs in terms of limits.
  • Discuss how understanding limits and continuity in multiple variables helps to analyze f(x,y) = sin(xy) in practical applications.
    • Understanding limits and continuity in functions like f(x,y) = sin(xy) is essential in many fields such as physics and engineering. For instance, when modeling wave behavior or oscillations influenced by two variables, knowing how these functions behave near critical points helps predict system responses. Analyzing continuity ensures reliable predictions without abrupt changes in outcomes. In real-world applications, having a grasp on these concepts allows for better design and stability in systems influenced by multidimensional variables.