I Flipping the sign in the definition of derivative

[Mr Davis 97]

Is it true that if $f$ is differentiable at $a$ that $f'(a) = \lim_{h\to a}\frac{f(a+h) - f(a)}{h} = \lim_{h\to a}\frac{f(a-h) - f(a)}{-h}$. That is, can the sign of $h$ be flipped. I've seen this a few times and it seems a bit dubious.

 

[mathwonk]

you mean h --> 0. and there is no difference between these two definitions, just changing the name of the variable from h to -h. to convince yourself use the trick for teaching algebra that uses an empty box instead of a variable, i.e. limit as ( )-->0, of f( a + ( ))/( ). and any letter you put in the box means the same thing, whether you call it h or s or -h or anything else. Well to be honest this is true because h-->0 if and only if -h -->0.

 

[fresh_42]

Differentiability means that we get the same derivative regardless whether we approach from the left or from the right.

 

[member 587159]

Is it true that if $f$ is differentiable at $a$ that $f'(a) = \lim_{h\to a}\frac{f(a+h) - f(a)}{h} = \lim_{h\to a}\frac{f(a-h) - f(a)}{-h}$. That is, can the sign of $h$ be flipped. I've seen this a few times and it seems a bit dubious.

If in doubt, $\epsilon-\delta$ is your best friend. Why don't you just try that?