If x and y are both postive

Question

If x and y are both positive , evaulte

lim p -> 0 (ln (o.3x^p+0.7y^p)/p =

and lim p-> 0 (0.3x^p +0.7y^p)^1/p=

Answer 1

First, note that lim_p→₀ (0.3x^p+0.7y^p )=1

Therefore lim[(0.3x^p +0.7y^p)^1/p]=lim_p→₀(1^1/p)=1

As to the 2sy function, ln(o.3x^p+0.7y^p)/p ,

because ln(0.3x^p+0.7y^p )⇒ln(1) when p⇒0 , we are dealing with an expression of 0/0 type when p→0

Therefore, use the rule

lim_p⇒₀[f(p)/g(p)]=lim_p⇒₀[f^'(p)/g^:(p)]. with f(p)=ln(0.3x^p+0.7y^p); g(p)=p

Because f'(p)=p(0.3x^p-1+0.7y^p-1)/(0.3x^p+0.7y^p) →0 when p→0 while g'(p)≡1 , lim_p⇒₀[f^'(p)/g^:(p)]=0