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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
1 year, 6 months 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

\[ \lim _ { h \rightarrow 0 } g ( x + bh ) \neq g(x)? \] Calvin Lin Staff · 1 year, 6 months ago

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@Calvin Lin 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? Dominick Hing · 1 year, 6 months ago

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@Dominick Hing 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? Calvin Lin Staff · 1 year, 6 months ago

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@Calvin Lin 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 Dominick Hing · 1 year, 6 months ago

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@Dominick Hing 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 \( f(x) = |x| \). Calvin Lin Staff · 1 year, 6 months ago

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@Calvin Lin Alright thanks I understand now! Dominick Hing · 1 year, 6 months ago

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