What if the limit is undefined

You tend to see " does not exist " when you encounter imaginary numbers in the context of real numbers, or perhaps when taking a limit at a point where you get a two-sided divergence, such as:.

This would be due to the fact that a limit does not exist when the limit from both the positive and negative direction differ it's like trying to make two north poles of magnets meet, and when they meet, if they meet, that is their limitbut they never meet. In those cases, either the limit from one side exists only, or the domain of the function does not contain the desired limit. Infinity is something that exists for us to quantify something that can never be truly reached in the absolute sense.

Infinity is just an arbitrarily large number that we attribute to solutions that we know will keep increasing or decreasing forever. The "final" value is then called oo , even though we never actually reach a final value. But we want to reach one, so we called it infinity. What's the difference between: undefined, does not exist and infinity? Truong-Son N. In this example the limit of f x , as x approaches zero, does not exist since, as x approaches zero, the values of the function get large without bound.

The values of the function "approach infinity", by which I mean that they get large and do not approach a real number. This is the function that represents the current flowing through the wire that connects the lamp on your desk to the wall socket.

Here the variable t is time. The lamp is initially turned off and at time zero you turn on the lamp so the current jumps from zero to one unit at that instant, and remains at that value until you turn off the lamp.

Here again the limit of f t , as t approaches zero does not exist. The reason this time is that if t approaches zero from the left side it seems that the limit should be zero but if t approaches zero from the right, it seems that the limit should be 1.

When such a situation arises the limit does not exist. As long as you treat this as a single limit, which is what you have done quite correctly, no problem arises.

In general, it is a false move to split a limit of a difference into the difference of two limits. It works sometimes, but then it needs to be justified carefully. On the other hand, if two limits are separately well defined, then their difference can be expressed as the limit of a difference assuming that the variable, domain, and endpoint of the limiting process is the same for both.

Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Can the difference of 2 undefined limits be defined? Ask Question. Asked 4 years ago. Active 3 years, 11 months ago. Viewed 2k times. Qudit 3, 4 4 gold badges 16 16 silver badges 31 31 bronze badges.

So you can't split the limit up here. In general, any limit law needs to be done with only finite limits involved. Certainly the limit of a difference can exist even if the limits of the terms being subtracted do not exist individually. If it were, it would definitely be undefined: arithmetic assumes definite operands. Add a comment. Active Oldest Votes. Andreas Blass Andreas Blass I think I'd steal your expression and "recycle" it for my students that blindly apply formulas without even thinking about the meaning of the variables they contain!

Alekos Robotis Alekos Robotis So the literal answer to your question is: The difference of 2 undefined limits cannot be defined, by definition. Community Bot 1.