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!

No vote yet

1 vote

×

Problem Loading...

Note Loading...

Set Loading...

## Comments

Sort by:

TopNewestThe second line is not valid. Can you think of a function such that at the point \( x \), we have

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

Log in to reply

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?

Log in to reply

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?

Log in to reply

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

Log in to reply

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 \( f(x) = |x| \).

Log in to reply

Log in to reply