Understanding Limits of Functions of Two Variables

What is the first step in evaluating limits of functions of two variables?

• Use algebraic techniques

• Perform direct substitution

• Check for indeterminate forms

• Graph the function

Why is direct substitution possible in the first limit problem?

• The denominator is zero

• The function is discontinuous

• There are no domain restrictions

• The numerator is zero

What is the result of the first limit problem after direct substitution?

• Infinity

• 0

• 1

• Undefined

What issue arises when performing direct substitution in the second limit problem?

• The function is continuous

• The result is an indeterminate form

• The numerator is zero

• The denominator is non-zero

What algebraic technique is used to simplify the second limit problem?

• Completing the square

• Substitution

• Factoring

• Expanding

What common factor is identified in the second limit problem?

• x^2 + y^2

• x^2 - y^2

• x^4 - y^4

• 2x^2 + 2y^2

After simplification, what is the new expression for the second limit problem?

• 6 * (x^2 + y^2)

• 6 * (x^2 - y^2)

• 3 * (x^2 - y^2)

• 3 * (x^2 + y^2)

What is the final value of the second limit problem after simplification?

• Undefined

• 3

• 6

• 0

What does the graph of the second limit problem show after simplification?

• A zero value

• An undefined point

• A continuous function

• A discontinuous function

What is the significance of the black point on the graph in the second limit problem?

• It confirms the limit value

• It shows a discontinuity

• It indicates an undefined point

• It represents a zero value