When can you take the limit of both sides?

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Homework Help Overview

The discussion revolves around the conditions under which limits, derivatives, integrals, absolute values, and supremum or infimum can be applied to both sides of an equation. The original poster seeks clarification on these mathematical operations and their validity in various contexts.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the conditions necessary for applying limits and other operations to both sides of an equation, questioning the continuity of functions involved and the implications of these operations.

Discussion Status

The conversation is ongoing, with some participants providing insights into the continuity of functions and how it affects the validity of operations. There is an acknowledgment that the applicability of these operations depends on the specific functions and context, indicating a productive exploration of the topic.

Contextual Notes

Participants note that the original poster's questions may not have straightforward answers and that the applicability of operations can vary significantly based on the functions in question.

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Homework Statement



A couple questions that have been building up here. I usually don't see them explicitly answered in the text.

When can you take the limit of both sides of an equation? And can you get rid of the limit from both sides of an equation?

How about taking the derivative or integral of both sides?

Can you ever take the absolute value of both sides?

What about taking the sup or inf of both sides?


These questions should help me a little bit.
 
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You can always do any of the above on both sides. Whether it exists on each side or whether they are equal on both sides depends completely on the problem. Sorry if that's not much an answer, but it's not a very good question.
 
Sorry about the question. I'm worse with words than math, and you know about my math skills Dick.

I see that a law like lim[ f(x) + g(x)] = f+g, might confirm this. I usually take this law in one direction -->. But going the other direction atleast partly justifies what I was asking.
 
If f and g are continuous at x0 then, sure, lim x->x0 f(x)+g(x)=f(x0)+g(x0) from either side. If they aren't then all bets are off. It depend exactly on what f and g are and what exactly you are trying to do. There is no one-size-fits-all answer to your question.
 

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