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# Calculus Limit Proof

Hi! I'm trying to prove that the following limit:

$$\lim _{ h\rightarrow 0 }{ \frac { f(x+bh+vh)-f(x+bh) }{ vh } } =\quad f'(x)$$

The way I am attempting to do this is by rewriting this as

$$\lim _{ h\rightarrow 0 }{ f'(x+bh) } =\quad f'(x)$$

I'm not sure if this is legal / valid. Could somebody give me some feedback and perhaps cite specific theorems, postulates ,etc? Thanks!

Note by Dominick Hing
2 years, 1 month ago

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The second line is not valid. Can you think of a function such that at the point $$x$$, we have

$\lim _ { h \rightarrow 0 } g ( x + bh ) \neq g(x)?$

Staff - 2 years, 1 month ago

The square root function. If $$x < 0$$ .

Let's say though that we defined $$f(x)$$ to be a polynomial of finite length (e.g. linear, quadratic, cubic, quartic, quintic, etc.). Would this then be a valid step?

- 2 years, 1 month ago

Well, the implicit assumption is that the function is defined in the domain, so try another counter-example.

Yes, it would work for polynomials. How do we know this is true?

Staff - 2 years, 1 month ago

The floor and ceiling functions, because the left side limit and right side limit are not equal if x is an integer...

About polynomials, they are continuous and differentiable across all real numbers

- 2 years, 1 month ago

Great!

Great! Polynomials are infinitely differentiable. For the 2nd statement to be true, what we needed was for $$f'$$ to be continuous. This would not hold always, like in the case of $$f(x) = |x|$$.

Staff - 2 years, 1 month ago

Alright thanks I understand now!

- 2 years, 1 month ago