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### Course: AP®︎/College Calculus BC>Unit 1

Lesson 3: Estimating limit values from graphs

# Unbounded limits

This video discusses estimating limit values from graphs, focusing on two functions: y = 1/x² and y = 1/x. For y = 1/x², the limit is unbounded as x approaches 0, since the function increases without bound. For y = 1/x, the limit doesn't exist as x approaches 0, since it's unbounded in opposite directions.

## Want to join the conversation?

• If I'm not mistaken, in an earlier video, when a function was approaching positive infinity from both left and right, the narrator/instructor still said that, since the limit was "unbounded," he would say that the "limit does not exist." (I believe he actually wrote it out that way.) I would be less anxious if he stuck with that label in these situations, or at least began the earlier video by saying that these situations can be and are described in different ways. Changing the definitions just a few videos later is a bit confusing.
• 'The limit does not exist' and 'the limit is unbounded' are not quite the same thing. If a limit is unbounded, then it does not exist. But a limit may not exist, and still be bounded, e.g. if we have a jump discontinuity.
• how do i simple this down?
• If the graph is approaching the same value from opposite directions, there is a limit. If the limit the graph is approaching is infinity, the limit is unbounded. A limit does not exist if the graph is approaching a different value from opposite directions.
• just curious but is there a shortcut or notation to write 'limit does not exist' when writing the limit?
• 3 years late, but yes. Sometimes people write "DNE" for "Does Not Exist."
• what is the value of an unbounded limit? what is it equal to?
• Nothing. If a function goes to infinity or negative infinity at a point, then the limit at that point doesn't exist.
• I'm confused why the limit of the first problem would not be infinity.
• because infinity is not something you can define
• Does unbounded situation counts as not exist?
• Yes. A limit is a real number that satisfies the ε-δ definition. Because infinity is not a real number, the limit doesn't exist when the function is unbounded.
• what if we use " +∞ , -∞ " to represent the limit is this correct for the first example ?
• So there is an infinite limit definition. Using that yes. (positive infinity)

I think the main goal would be to describe the behaviour of a function as much as possible. In the video, Sal is using the standard epsilon delta definition of limits which if used would result in the answer undefined.

So the definition he is using will not describe the function behaviour in detail but he makes note of that in the video.

It is not meaningful to use the standard epsilon delta definition in this video. Limits are essentially are combinations of definition, standard epsilon delta, infinite limits, limits at infinity, one-sided limits.

It is by convention in mathematics to use limit definition that describe the function in the most detail. Hence it is best to use the infinite limit definition in this scenario.
• Could +∞ be an answer to the first question?
• For this case, you could. But, know that unbounded limit does not always equal an infinite limit.

For example, for \$f(x) = xsin(x)\$, as \$x\$ tends to infinity, \$f(x)\$ is unbounded, but doesn't tend to infinity. It oscillates between multiple values.