Understanding Limits: Real Numbers or Infinity?

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The discussion centers on the concept of limits in calculus, specifically addressing whether a limit can equal infinity. It is established that while infinity is not a real number, in the context of limits, it is common to state that the limit approaches infinity rather than being undefined. For instance, the limit of 1/x as x approaches 0 from the right is considered to be infinity in the extended real number system, which includes infinity. This shorthand is widely accepted for clarity in mathematical communication.

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  • Understanding of calculus concepts, particularly limits
  • Familiarity with real numbers and extended real numbers
  • Knowledge of functions and their behaviors near critical points
  • Basic mathematical notation and terminology
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  • Study the properties of limits in calculus
  • Explore the extended real number system and its implications
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I can't believe I'm asking this, because I should know this answer but I'm now doubting myself.

If a function goes to infinity as x approaches some real number, would we say the limit as x approaches that number is infinity or would we say that it does not exist?

Doesn't the limit always have to be a real number, but infinity isn't one?

So for example:
the limit as x approaches 0 from the right of 1/x is infinity or not defined?
 
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It is a question of whether you are in the reals or the extended reals. The latter includes infinity.
 
Generically people will say the limit equals infinity. Indeed infinity is not a real number, but it is useful to use the shorthand of saying the limit equals infinity, as this is more specific than just saying that the limit is undefined.
 

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