Question

Let's compute the given limits:

a) lim(x,y)→(4,2) cos(πxy)−sin(x−y2) / x²+y²
b) lim(x,y)→(0,0) x²+y² / √1−cosx² +y2

Answer 1

The limit of the expression cos(πxy)−sin(x−y/2)/(x²+y²) as (x,y) approaches (4,2) is 1. The limit of the expression (x²+y²)/(√(1−cos(x²+y²))) as (x,y) approaches (0,0) is 0.

Step-by-step explanation:

To compute the given limits:

a) To find the limit of the expression cos(πxy)−sin(x−y/2)/(x²+y²) as (x,y) approaches (4,2), we substitute the values into the expression to get cos(8π)−sin(2−1/2)/(16+4), which simplifies to 1−0/20, resulting in a limit of 1.

b) To find the limit of the expression (x²+y²)/(√(1−cos(x²+y²))) as (x,y) approaches (0,0), we substitute the values into the expression to get 0²+0²/(√(1−cos(0²+0²))), which simplifies to 0/√(1−1), resulting in a limit of 0.