Explain what it means to say that   `lim_(x->1^-)f(x) = 3` ` ` `lim_(x->1^+) f(x) = 7` In this situation is it possible that the limit as x-> 1 exists? Explain. `` ` `

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embizze eNotes educator| Certified Educator


This is the limit of a function as the input x approaches 1 from the left. Technically, this means that for any real number epsilon greater than 0, there is a delta greater than 0 such that the distance from the value of the function from 3 is less than epsilon whenever x is less than delta away from 1 from the left. Naively this means that the function gets arbitrarily close to 3 as x approaches 1 from the left.``

Similarly, the other limit is from the right.

In this situation, the limit as x approaches 1 does not exist. For the limit to exist, the left hand and right hand limits must exist and agree.

Here is a possible graph of a function with the given limits:


The limit as x approaches 1 of such a function does not exist.