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So let's say you have to find limits...
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So let's say you have to find limits using only algebraic manipulation. No effective methods like applying epsilon delta or using l'Hopital are allowed.
How do you know whether the limit you arrived at using algebraic manipulation is correct? One can get different results via different algebraic manipulations. How do you know that it's time to plug in the limit and just calculate?
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The only was I see a purely algebraic way of finding a limit would work if it ends with

limt_{h -> 0} const. = const.

as in the limit h to 0 of

$\frac {(x+h)^2-x^2} {x+h} = \frac {x^2 + 2xh +h^2 -x^2} {x+h} = 2x \frac {x+h} {x+h} =2x$

In that case the solution will be unique, yes.
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>>7858022
Could we perhaps work with preimages and open sets?
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Dumb weeaboo
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>>7858022
>One can get different results via different algebraic manipulations
wut
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>One can get different results via different algebraic manipulations.
In a Hausdorff space every limit is unique.
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>>7858126
Why are we assuming that we are even in Hausdorff space?
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>>7858114
How do I know when I should plug the x in when evaluating a limit, after algebraically manipulating it?

If I plug the x in after barely doing any manipulation, I get one result (usually undefined).
If I plug it in after a lot of manipulation, I get another one.
How do I know which one is correct?
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>>7858134
Also, after doing even more manipulation, the result may be entirely different after I plug the x in.
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>>7858134
[eqn] \lim_{x \to c} f(x) = f(c) [/eqn]
Is only true is $f$ is continuous in $c$.
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>>7858060
How did you get from step 2 to step 3?
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>>7858133
Why would we make crazy assumptions like that?
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>>7858134
>How do I know when I should plug the x in when evaluating a limit,
When you're able to. If you have 0/0 then the form is indeterminate, you need to do more work. Note that an indeterminate form does not necessarily mean the limit does not exist.
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>>7858184
Because there are plenty of easy to think about topologies which we can experiment with.
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>>7858134
Can you please differentiate between undefined and indeterminate?
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>>7858179
I made an error, the denominator must be h, not x+h.
Result is the same.