Derivative Zero-Function

[Guest]

Hello,
it's my first time here. Thanks for helping.

I came across this:

lim.png (42.85 KiB) Viewed 652 times

My question is about the last line. Is it allowed to simplify the term to 0? Or is this a mistake.
For me it makes sense, cause if I try to devided nothing to something it will be nothing for everyone.
But another math teacher was not so sure.

What's you opinion?
Thanks

[Guest]

Yes! 0 divided by any number (except 0) is 0. In fact, if f(x) is any constant, not just 0, by the same argument, has derivative 0.

[Guest]

[tex]\frac{a}{b}= c[/tex] is equivalent to [tex]a= bc[/tex] so [tex]\frac{0}{b}= c[/tex] is equivalent to [tex]0= bc[/tex]. Since b is not 0, c must be 0 for that to be true. (b cannot be 0 because [tex]\frac{0}{0}[/tex]is "indeterminate".)

[Guest]

Yes, 0 divided by any non-zero number is 0. That's pretty much basic arithmetic.