Question

☺ Chee Check who whether f(x) = { sinx fx to it differentiable at Pool r = 0 x & proof

Answer 1

We have f(x)= Sina nto x if of x=0 Now, we know, f'(o) exists exists i e, fis differentiable at x20 iff the limit lim f(x) - f (1) exists ato e-0 Now, lim f(w) - f (0) lim Sinn -1 NO x x-0 Mto _

We noi use some inequalities. we know, the following inequalities. are true for values close to zero both positive and negetive. | Sinal <ixl and kal < Hanxl This in turn implies that 1 Cos*/ < Sinx) < x Since; x is close to zeno (x+0)

So, here we will take the interval. 61, ) and this implies. Cosa < Sinx <l x Cosa-l sinx-l.co. X Taking absolute values, we get Sinx -11 < 11- Cosul 0 Now, I |- Cosal Sinza-l<x? |- Costne It Cos a It Cosa [ . : It Cost > wheneven. & is close to zero, both positive and negetive and sinal alal. =) Sinha < x2 2 거 ) 11-Cosze 11-Cosz | < 1224=161.104] |- Cosa <1%] tence, from ③ we have 9 x. Sina 2 < llor cosul Cosul <lxl Ixl

Let E70 then Sina. <E. 7 x whenever laics (=E). So, t Exo thene exist. S=)0 such that sinx-l les 2 wheneveni CE al < s <=6). le, lim Sina | exists and.. M 10 x equals to D. lim f(n)-f(0) exists and equals no x-0 to O. : f'(o) exists and f'(o)=0. i. e, f (1) is differentiable at .x=0

Hope this will help you. For any further doubt please comment below. Thank you.