Limits and Asymptotes: A Deep Dive

What does it signify when a function tends towards positive or negative infinity as x approaches a finite value?

• The function is continuous.

• The function has a horizontal asymptote.

• The function is differentiable.

• The function has a vertical asymptote.

In the example of the function 1/(x-1)^2, what happens to the y-values as x approaches 1 from either direction?

• They approach a finite number.

• They oscillate between positive and negative values.

• They tend towards infinity.

• They remain constant.

What is the more nuanced interpretation of the limit of a function that tends towards infinity?

• The limit is a finite number.

• The limit does not exist.

• The limit is zero.

• The limit is infinity.

What is indicated by the presence of an infinite limit?

• A horizontal asymptote.

• A vertical asymptote.

• A point of discontinuity.

• A point of inflection.

When evaluating one-sided limits, what conclusion can be drawn if the limits from the left and right do not match?

• The overall limit exists.

• The overall limit does not exist.

• The function is differentiable.

• The function is continuous.

What does it signify when a function approaches a finite limit as x approaches plus or minus infinity?

• The function has a vertical asymptote.

• The function has a horizontal asymptote.

• The function is continuous.

• The function is differentiable.

In the example of the function y = e^x, what happens to the y-values as x approaches infinity?

• They approach zero.

• They tend towards negative infinity.

• They tend towards infinity.

• They remain constant.

What is the limit of the function 2x^3 / 5x^3 as x approaches infinity?

• 0

• Infinity

• 1

• 2/5

What happens to the limit of the function 4 / (x-3)^2 as x approaches 3 from either direction?

• It does not exist.

• It tends towards negative infinity.

• It tends towards positive infinity.

• It tends towards zero.

What is the limit of the function e^(1/(x-2)) as x approaches 2 from the right?

• One

• Negative infinity

• Zero

• Infinity