Finding the limit of (SiNx) / X when limx tends to zero

Finding the limit of (SiNx) / X when limx tends to zero

im (x->0) sin(sinx)/x
=lim (x->0) [sin(sinx)/sinx] * [sinx/x]
∵x->0 ; t= sinx-> 0,
lim (x->0) [sin(sinx)/sinx] = lim (t->0) sint/t = 1
=1*1
=1

How to find limx →∞ (x / x-1) ^ x limit?

The limit limx tends to 0 (1 / e ^ x-1) - (1 / x)

By Using Equivalent Infinitesimal Substitution and lobita's rule, the original formula = LIM (x → 0) (x-e ^ x + 1) / (x (e ^ x-1)) = LIM (x → 0) (x-e ^ x + 1) / x ^ 2 = LIM (x → 0) (1-e ^ x) / (2x) = - 1 / 2lim (x → 0) (e ^ x-1) / x = - 1 / 2