Why Some Rational Functions Have No Vertical Asymptote

The video tutorial explores rational expressions that do not have vertical asymptotes, addressing common student misconceptions. It provides three examples: simplifying rational expressions, identifying removable discontinuities, and using complex numbers to explain the absence of vertical asymptotes. The tutorial emphasizes the importance of understanding when a function is undefined and how this relates to vertical asymptotes.

What is a vertical asymptote in the context of rational expressions?

• A point where the graph crosses the y-axis

• A point where the graph crosses the x-axis

• A value of x that makes the denominator zero, causing the function to be undefined

• A value of x that makes the numerator zero

In the first example, why is there no vertical asymptote?

• The numerator is zero

• The function is not a rational expression

• The graph crosses the x-axis

• The denominator is a constant and never zero

What is a removable discontinuity?

• A point where the graph crosses the y-axis

• A value that makes the function undefined but can be simplified out

• A point where the graph crosses the x-axis

• A value that makes the numerator zero

In the second example, what happens to the vertical asymptote after simplification?

• It remains a vertical asymptote

• It disappears completely

• It becomes a horizontal asymptote

• It becomes a hole in the graph

Why does the third example not have a vertical asymptote?

• The function is not a rational expression

• The graph crosses the x-axis

• The denominator has no real solutions that make it zero

• The numerator is zero

What mathematical concept is introduced in the third example to explain the absence of vertical asymptotes?

• Imaginary numbers

• Complex numbers

• Real numbers

• Rational numbers

What must be true for a vertical asymptote to exist in a rational expression?

• The numerator must be zero

• The function must be linear

• The denominator must be zero with real solutions

• The graph must cross the y-axis