Question

what is the value of limit x tends to 0, (sin nx-nsinx)/x(cos nx-cosx)​

Answer 1

$sinx = x -x^3/3! +x^5/5! -x^7/7! +…(sinx)/x = 1 - x^2/3! +x^4/5! -x^6/7! +…$

Clearly lim, x—->0 of $(sinx)/x$ = 1.

Then lim, x—->0 of $((sinx)/x)^((sinx)/x -sinx)) = 1^(1–0) = 1^1 = 1.$