For each of the following functions f(0)=0. Determine the derivative at 0 or else show that f'0 does not exist

Question

i) f(x)=x^2sin(1/x)

ii) f(x)=xe^1/x

iii) f(x)=x/sin(x^2)

Answer 1

Just plug ahead. For example, on #3, using thproduct rule gives f' = e^(1/x) + x * e (1/x) * (-1/x^2)

The first term has limit 1.

The second is a lot like the limit as y goes to infinity of e^y/y, which blows up (substitute y = 1/x). So the whole thing blows up.

The key here is that by looking for f'(0) what you're really looking for is the limit as x goes to 0 of f'(x).

Answer 2

we are able to teach via induction that the nth by-fabricated from e^(-a million/x^2) has the type g_n(x) * e^(-a million/x^2) the place g_n(x) is a rational function. Then we use the decrease definition of derivative, and induction on n. If f^(n)(x) exists for all x, then f^(n+a million)(0) = lim x->0 f^(n)(x)/x = lim x->0 g_n(x)*e^(-a million/x^2)/x. This decrease is 0 because of the fact e^(-a million/x^2) techniques 0 swifter than the denominator of g_n(x). EDIT: Oops, John D published a greater useful answer together as i became into composing mine.