Understanding Limits of Functions of Two Variables

The video tutorial covers the concept of limits for functions of one and two variables. It explains how limits are determined by approaching a point from different paths and discusses the importance of continuity. Several examples illustrate how to find limits at specific points, including cases with discontinuity and the use of direct substitution. The tutorial also demonstrates simplifying rational functions to eliminate discontinuities and determine limits.

What is required for the limit of a function of two variables to exist at a point?

• The limit must be infinite from all paths approaching the point.

• The limit must be zero from all paths approaching the point.

• The limit must be different from all paths approaching the point.

• The limit must be the same from all paths approaching the point.

When can direct substitution be used to find the limit of a function of two variables?

• When the function is discontinuous at the point.

• When the function is linear.

• When the function is continuous around the point.

• When the function is undefined at the point.

What happens if a function is not continuous at a point?

• The limit does not exist.

• The limit is always zero.

• The limit can be determined by direct substitution.

• The limit must be considered from different paths.

In the example where the limit as XY approaches (1, 1), what was the result of direct substitution?

• The limit was 3.

• The limit was 2.

• The limit was 5.

• The limit was 4.

What is the significance of the line Y = -X in the example discussed?

• It indicates a point of continuity.

• It shows a path where the limit is zero.

• It represents a path where the function is discontinuous.

• It is a path where the limit is infinite.

What is the result when approaching the origin along the path Y = X?

• The limit is 1/2.

• The limit is 0.

• The limit is 1.

• The limit is -1/2.

What conclusion can be drawn if limits from different paths are not the same?

• The limit is infinite.

• The limit is zero.

• The limit does not exist.

• The limit exists.

How can a rational function be simplified to determine a limit?

• By multiplying by zero.

• By dividing by zero.

• By adding a constant.

• By factoring and canceling common terms.

What is the limit as XY approaches (2, 4) after simplifying the rational function?

• The limit is 4.

• The limit is 3.

• The limit is 5.

• The limit is 2.

What happens to the limit of a function as XY approaches (0, 0) if the surface goes down quickly?

• The limit is zero.

• The limit is undefined.

• The limit is negative infinity.

• The limit is positive infinity.