Question

sinx 5. (10 marks) Check whether f(x) = if x 70 if x = 0 is differentiable at x=0. Prove it using an e-8 proof.

Answer 1

f(x) sinx if xto 1 if x=0 solution;- of f is differentiable at xa with derivative pica) if for any E>0, there exist s such that x ER o<lx-alus for any - f(x) fla) xe - a fica) ul have I By Taylor's series f(x) = sinx 43 x SI fo) = sinn x2 3! X 7! s! lim fix tim Bina) =lim + X+0 Xto x>0 53 1 lim Xo as f(o)

Hence, f(x) continuous at Now, f'(x) [x=0 0 1 1 flo ALSO, f(x) - f(a) x-a - fla) at a-0, we have Sinu 1. 0 0 sinx - x 0 O x2 form (By L'Hospital's Rule coon-1 lim no = lim xto sina 2 c LE This function is differentiable at x=0