Question

Is lim (h -> 0) of h*f(x) = 0 always true?

Answer 1

where is the function?

Answer 2

function is R to R and differentiable that the only thing I know

Answer 3

but isn't 0 * anything = 0 ? @magepker728

Answer 4

what is hf(x)?

Answer 5

or u mean f(h)?

Answer 6

@magepker728 h*f(x)

Answer 7

i think its true.

Answer 8

https://www.wolframalpha.com/input/?i=is+lim+(h+-%3E+0)+of+h*f(x)+%3D+0+always+true

wolfram thinks so

Answer 9

Assuming f nor x are functions of h, then it has to be true, since the limit would then treat f(x) as a constant. But this is not necessarily the case if either f or x is a function of h.
For example:
Assuming \(x \neq 0\), if \(f(x)= \frac x h\) or if \(f(x)=x\) and \(x=\frac 1 h\), then \(\lim(hf(x))=\lim(\frac {hx} h)=\lim x=x \neq 0\)