If `underset(xtoa)lim(a^(x)-x^(a))/(x^(x)-a^(a))=-1" and "agt0," then find the value of a."`

If lim_(xtoa)(a^(x)-x^(a))/(x^(x)-a^(a))=-1" and "agt0," then find the value of a."

underset (xrarr0)lim x^(x)

underset(xrarr0)lim((1)/(sin x)-(1)/(x))

underset(xrarr0)lim (cos x) /(x^(2))

Evaluate: underset(xrarr0)lim((log(1-x))/((x)))

lim_(xtoa) (log(x-a))/(log(e^(x)-e^(a)))

Evaluate: underset(xrarr0)lim((1)/(x)-(1)/(e^(x)-1))

Let f(a)=g(a)=k and their nth derivatives exist and be not equal for some n. If lim_(xtoa) (f(a)g(x)-f(a)-g(a)f(x)+g(a))/(g(x)-f(x))=4 then find the value of k.