Exploring Infinite Limits and Vertical Asymptotes

What is a vertical asymptote?

• A point where the graph crosses the x-axis

• A point where the graph crosses the y-axis

• A line where the graph intersects at multiple points

• A line where the graph approaches but never touches

When does a vertical asymptote occur?

• When the graph has a hole

• When the numerator equals zero

• When the denominator equals zero and the factors do not cancel

• When both numerator and denominator equal zero

What happens to the graph as it approaches a vertical asymptote?

• It remains constant

• It oscillates between positive and negative values

• It approaches a finite value

• It approaches either positive or negative infinity

If the limit as x approaches 3 from the left is negative infinity, what does this indicate about the graph?

• The graph has a hole at x=3

• The graph is constant

• The graph is decreasing towards negative infinity

• The graph is increasing towards positive infinity

How can you determine the behavior of the graph near a vertical asymptote without a graph?

• By using a graphing calculator

• By solving for x-intercepts

• By plugging in values close to the asymptote

• By finding the derivative

What is the limit of f(x) as x approaches 3 if the left and right limits are not the same?

• The limit exists and is finite

• The limit does not exist

• The limit is zero

• The limit is positive infinity

In the example where x approaches 1 from the right, what is the approximate value of the numerator?

• -2.001

• 1.999

• -1.999

• 0.001

Why is the denominator squared in the example where x approaches 1?

• Because it represents a hole

• Because it cancels out the numerator

• Because it simplifies the calculation

• Because there are two identical factors in the denominator

What is the limit of f(x) as x approaches 1 from both sides in the detailed example?

• Positive infinity

• Does not exist

• Negative infinity

• Zero

What should you do if the left and right side limits are different?

• Average the two limits

• Take the limit from the left side

• Take the limit from the right side

• State that the limit does not exist