Understanding One-Sided Limits and Vertical Asymptotes

The video tutorial explains the concept of one-sided limits and their relationship with vertical asymptotes. It describes how a vertical line x = a is a vertical asymptote if the limit of f(x) as x approaches 'a' from either side equals positive or negative infinity. The tutorial provides two examples: one where x approaches 2 from the right, leading to positive infinity, and another where x approaches 2 from the left, leading to negative infinity. Both examples demonstrate that x = 2 is a vertical asymptote.

What is a vertical asymptote in the context of one-sided limits?

• A line that the graph intersects at infinity

• A point where the graph crosses the x-axis

• A line that the graph approaches but never crosses

• A point where the graph crosses the y-axis

When does a vertical line x = a become a vertical asymptote for a function f(x)?

• When the limit of f(x) as x approaches a is +/- infinity

• When the limit of f(x) as x approaches a is undefined

• When the limit of f(x) as x approaches a is a finite number

• When the limit of f(x) as x approaches a is zero

In the first example, what happens to the function values as x approaches 2 from the right?

• They decrease without bound

• They remain constant

• They increase without bound

• They oscillate

Why does the limit not exist as x approaches 2 from the right in the first example?

• Because the function values approach negative infinity

• Because the function values approach positive infinity

• Because the function values remain constant

• Because the function values approach zero

What is the behavior of the denominator as x approaches 2 from the right?

• It remains constant

• It approaches a large positive number

• It approaches zero and is always negative

• It approaches zero and is always positive

In the second example, what happens to the function values as x approaches 2 from the left?

• They oscillate

• They decrease without bound

• They remain constant

• They increase without bound

Why does the limit not exist as x approaches 2 from the left in the second example?

• Because the function values remain constant

• Because the function values approach negative infinity

• Because the function values approach positive infinity

• Because the function values approach zero

What is the behavior of the denominator as x approaches 2 from the left?

• It approaches a large positive number

• It approaches zero and is always positive

• It approaches zero and is always negative

• It remains constant

How does the function behave when dividing by a very small negative number?

• It approaches negative infinity

• It approaches zero

• It remains constant

• It approaches positive infinity

What does the table of values demonstrate in the second example?

• The function values oscillate

• The function values approach negative infinity

• The function values approach positive infinity

• The function values remain constant