Can Terms Always Be Canceled Out in Math?

[wajed]

I thought we can always cancel terms out and that its absolutely acceptable
but then it turned out that I can`t cancel out the term "x-2" from "(x^2 - 4)/(x-2)"
is this exception only for functions?
and I have a 2nd question, and I know I should read on logic first, but PLEASE! I want to know this:>
If someone proved that canceling terms out is ok, then how come we face some situations where we just can`t cancel the terms out without implying that we did so?


My 3rd question, why does canceling terms out is ok without implication when we deal with limits while its not when we deal with functions?



Thank You

 

[yyat]

The only difference between (x^2-4)/(x-2) and x+2 is that the former is not defined for x=2. As long as you are aware of that, regarding them as the same function is okay.

When computing limits, for example, they can be interchanged.

 

[wajed]

"My 3rd question, why does canceling terms out is ok without implication when we deal with limits while its not when we deal with functions?"

Can anyone answer this one?

 

[matt grime]

This looks like one of those annoying semantic arguments about functions, which are commonly misdefined in calc classes, sadly. As such I'm going to have to say something that I disagree with.

You can cancel things whenever they are not zero - in those limits you're actually thinking about the ratio of two real non-zero quantities in an attempt to work out how you ought to define something like

(x^2 - 4)/(x - 2)when we would like to substitute x=2 into that expression, but can't because of a 0/0 issue.

Of course, since everything in sight is continuous you can just divide out the denominator from the numerator and get the same answer.

Essentially you should divide then take a limit, not take a limit then divide.

 

[John Creighto]

My response was going to be, "I'm okay with it". Anyway, everything depends on context, if you were designing a computer algebra system for instance, you may want to deal with separate cases where x is not equal to 2 and x is equal to 2.

The question is how did we arrive at the expression (x^2 - 4)/(x - 2). If we got their by dividing some equation by (x-2) then I'm sure we can find some cases wear assuming that cancellation is okay will lead to erroneous results.

Anyway, I think that the cancellation is normally safe to do but if a computer algebra system simplifies the expression without us telling it to do the cancellation, it should list the assumptions it made. Maybe x=2 might be very meaningful with regards to the application.

 

[HallsofIvy]

"My 3rd question, why does canceling terms out is ok without implication when we deal with limits while its not when we deal with functions?"

Can anyone answer this one?

Definition: $\lim_{x\rightarrow a} f(x)= L$ if and only if, given any $\epsilon$> 0 there exist a $\delta> 0$ such that if $0< |x-a|< \delta$, then $|f(x)- L|< \delta$.

Let us say that both numerator and denominator have a factor of x-a. If you are taking a limit as x goes to b, not equal to a, then there is no problem. No matter how close to a b is, the limit depends only on values even closer (the $\delta$ in an "$\epsilon$, $\delta$" proof can be taken smaller) so we can avoid a. If you are taking the limit as x goes to a, there is still no problem: the $0< |x-a|$ part in the definition above means we never have to look at x= a.

People are telling you that you can cancel the "x- a" terms as long as x is not equal to a and, working with limits we can always avoid any single value.

 

[wajed]

Thank you all,

and, not trying to be rude, but last answer was the best, it was perfect, and I even learned things that are more than just an answer to the question.

Thank you all again