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!

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TopNewestThe 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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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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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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About polynomials, they are continuous and differentiable across all real numbers – Dominick Hing · 1 year, 6 months ago

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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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– Dominick Hing · 1 year, 6 months ago

Alright thanks I understand now!Log in to reply