Understanding One-Sided Limits and Vertical Asymptotes

The video tutorial explains the concept of one-sided limits and their relationship with vertical asymptotes. It provides examples of limits as x approaches 2/3 from both the right and left sides, demonstrating how the function behaves and identifying vertical asymptotes. The tutorial concludes with an algebraic method to find vertical asymptotes by setting the denominator to zero.

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

• A point where the graph crosses the y-axis

• A line the graph approaches but never crosses

• A line the graph intersects at infinity

• A point where the graph crosses the x-axis

What happens to the function values as x approaches 2/3 from the right in the given example?

• They increase without bound

• They decrease without bound

• They oscillate

• They remain constant

Why does direct substitution not work for the limit as x approaches 2/3 from the right?

• The numerator becomes zero

• The denominator becomes zero

• The function is undefined

• The limit is infinite

What does the table of values show as x approaches 2/3 from the right?

• Function values oscillate

• Function values remain constant

• Function values decrease without bound

• Function values increase without bound

What is the limit as x approaches 2/3 from the left?

• Zero

• Undefined

• Negative infinity

• Positive infinity

What does the left-side limit tell us about the vertical asymptote at x = 2/3?

• It disproves the asymptote

• It suggests a horizontal asymptote

• It indicates no asymptote

• It confirms the asymptote

How do function values behave as x approaches 2/3 from the left?

• They decrease without bound

• They increase without bound

• They remain constant

• They oscillate

What should be checked in rational functions to find vertical asymptotes?

• Neither numerator nor denominator equals zero

• Denominator equals zero

• Numerator equals zero

• Both numerator and denominator equal zero

What equation is derived to find the vertical asymptote in the example?

• 3x + 2 = 0

• 2 - 3x = 0

• 2 + 3x = 0

• 3x - 2 = 0

What is the final step to solve for x in the vertical asymptote equation?

• Add 2 to both sides

• Subtract 2 from both sides

• Divide both sides by -3

• Multiply both sides by 3