Is it okay to divide by variables in calculus?

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Dividing by variables in calculus is permissible when the variable is not equal to zero, as demonstrated in the function Q(h) = (2h + h^2)/h. The discussion clarifies that this operation simplifies the expression to Q(h) = 2 + h without eliminating solutions, since the limit as h approaches zero is considered. The confusion arises from the general rule against dividing by variables in equations, which can lead to loss of solutions. In this context, the focus is on simplification rather than solving an equation, making the division valid. Understanding this distinction is crucial for applying calculus concepts correctly.
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[SOLVED] Why eliminate answers?

In my book, Essential Calculus, a section is introduced with limits. They introduce a random(maybe) function :

Q(h) = \frac {2h + h^2}{h} (h \neq 0)

"We then divide the numerator by the denominator h, which is permissible since h \neq 0. This gives the simple formula Q(h) = 2 + h (h \neq 0)"

I was always told you do not divide by variables like in a trigonometric equation because it eliminates solutions. How are they then to say that it is permissible since h \neq 0 ? I don't understand.
 
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In this case, it is fine. Given that h \neq 0, both functions are equal because you are only simplifying the function.
 
razored said:
I was always told you do not divide by variables like in a trigonometric equation because it eliminates solutions. How are they then to say that it is permissible since h \neq 0 ? I don't understand.

Hi razored! :smile:

It's ok because all these calculus equations begin "lim as h -> 0".

So it's impossible for h to be 0. :smile:

(oh … and look up L'Hôpital's Rule :wink:)
 
razored said:
In my book, Essential Calculus, a section is introduced with limits. They introduce a random(maybe) function :

Q(h) = \frac {2h + h^2}{h} (h \neq 0)

"We then divide the numerator by the denominator h, which is permissible since h \neq 0. This gives the simple formula Q(h) = 2 + h (h \neq 0)"

I was always told you do not divide by variables like in a trigonometric equation because it eliminates solutions. How are they then to say that it is permissible since h \neq 0 ? I don't understand.
What "solutions" do you mean? You are not "solving" an equation here.
 
HallsofIvy said:
What "solutions" do you mean? You are not "solving" an equation here.
Whoops. That is what I misunderstood. Thanks for pointing that out!
 

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