Question

For each of the functions below say whether the limit exists at the point (0,0). If the limit exists then find it.

f₁(x,y)= (x²sinx−ysiny+ysinx−x²siny)/ (y+x²)

Answer 1

To determine whether the limit exists at (0,0) for the function f₁(x,y), we can simplify the function by factoring out common terms, approach the limit from different paths, and evaluate the limit along each path.

Step-by-step explanation:

To determine whether the limit exists at the point (0,0) for the function f₁(x,y) = (x²sinx−ysiny+ysinx−x²siny) / (y+x²), we need to evaluate the limit as (x,y) approaches (0,0).

To do this, we can use the limit laws and techniques. First, we can simplify the function by factoring out common terms, resulting in f₁(x,y) = (x(xsinx−ysiny)+y(ysinx−x²siny)) / (y+x²).

Next, we can try approaching the limit from different paths. For example, if we approach along the x-axis (y=0), the function becomes f₁(x,0) = (x(xsinx)+0) / (0+x²) = xsinx/x² = sinx/x.

As x approaches 0, sinx/x approaches 1 (using a well-known limit), so the limit exists and is equal to 1. Similarly, we can evaluate the limit along the y-axis (x=0) and other paths to verify that the limit exists and is equal to 1 at (0,0).