Understanding Limits and Vertical Asymptotes

The video tutorial explains the concept of limits, focusing on one-sided limits and their connection to vertical asymptotes. It examines specific examples involving the cosecant function, demonstrating how limits behave as X approaches 0, Pi, and 3PI/2. The tutorial uses graphs and tables to illustrate how function values approach infinity or specific values, confirming the existence or non-existence of limits. The video aims to enhance understanding of limits and asymptotes in mathematical functions.

What is a vertical asymptote in the context of limits?

• A line that the graph crosses at infinity.

• A point where the graph intersects the x-axis.

• A vertical line the graph approaches but never crosses.

• A horizontal line the graph approaches but never crosses.

When does a vertical asymptote occur for a function f(x) at x = a?

• When the derivative of the function is zero at x = a.

• When the function is continuous at x = a.

• When the limit as x approaches a from the left or right equals infinity.

• When the limit as x approaches a from both sides equals zero.

What happens to the function values as x approaches 0 from the left for the cosecant function?

• They decrease without bound.

• They increase without bound.

• They remain constant.

• They approach zero.

Why does the limit as x approaches 0 from the left for cosecant x not exist?

• Because the function values approach zero.

• Because the function values approach positive infinity.

• Because the function is continuous at x = 0.

• Because the function values approach negative infinity.

What is the significance of x = 0 for the cosecant function?

• It is a point of intersection with the x-axis.

• It is a vertical asymptote.

• It is a horizontal asymptote.

• It is a point of continuity.

As x approaches π from the right for the cosecant function, what do the function values do?

• They increase without bound.

• They remain constant.

• They decrease without bound.

• They approach zero.

What does it mean if the limit as x approaches π from the right equals negative infinity?

• The function has a horizontal asymptote at x = π.

• The limit does not exist.

• The limit exists and is finite.

• The function is continuous at x = π.

What is the limit of cosecant x as x approaches 3π/2 radians?

• -1

• 0

• Infinity

• 1

Why can direct substitution be used for the limit as x approaches 3π/2 radians for cosecant x?

• Because the function is discontinuous around this value.

• Because the function is undefined at this value.

• Because the function has a vertical asymptote at this value.

• Because the function is continuous around this value.

What is the function value of cosecant at 3π/2 radians?

• 0

• -1

• 1

• Infinity