Understanding Limits in Multivariable Calculus

The video tutorial explores the concept of limits in multivariable calculus by examining the limit of the function 6xy / (9x^2 + y^2) as (x, y) approaches (0, 0) along two different paths: y = x and y = -2x. Through graphical analysis and calculations, it is demonstrated that the limits along these paths are different, leading to the conclusion that the general limit does not exist. The tutorial emphasizes the importance of approaching the same function value from all paths for a general limit to exist.

What is the function whose limit we are trying to find as (x, y) approaches (0, 0)?

• 9xy / (6x^2 + y^2)

• 9x^2 / (6xy + y^2)

• 6x^2 / (9xy + y^2)

• 6xy / (9x^2 + y^2)

What must be true for a general limit to exist in multivariable calculus?

• The limit must be the same from all paths.

• The limit must be undefined.

• The limit must be zero.

• The limit must be infinite.

What is the significance of the graph in three dimensions in this context?

• It proves the function is continuous.

• It helps visualize the different paths and their limits.

• It shows the function is linear.

• It demonstrates the function is quadratic.

What substitution is made to find the limit along the path y = x?

• Substitute x = -y

• Substitute y = 2x

• Substitute x = y

• Substitute y = x

What is the simplified limit along the path y = x?

• 1/2

• 6/10

• 3/5

• 2/3

What is the function value along the path y = x as we approach (0, 0)?

• Below the XY plane

• Undefined

• Above the XY plane

• On the XY plane

What substitution is made to find the limit along the path y = -2x?

• Substitute y = -2x

• Substitute y = 2x

• Substitute x = -2y

• Substitute x = 2y

What is the simplified limit along the path y = -2x?

• -12/13

• -6/10

• -3/5

• -1/2

Why does the general limit not exist for the given function?

• The function is not continuous.

• The function has no maximum value.

• The limits along different paths are not equal.

• The function is undefined at (0, 0).

What is the function value along the path y = -2x as we approach (0, 0)?

• Below the XY plane

• On the XY plane

• Undefined

• Above the XY plane