Given:

underset \((x->5^+)(lim)(1/x^(4/3)-1/(x-5)^(4/3))\)

It is known that

underset \((x->a)(lim)[f(x)+-g(x)]=underset(x->a)(lim)f(x)+-underset (x->a)(lim)g(x)\)

Then above can be written as

\(=underset(x->5+)(lim)(1/x^(4/3))-underset(x->5+)(lim)(1/(x-5)^4/3)\)

Plug Limits

\((1/5^4/3)-(oo)(since,(x-5)^(4/3)>0)\)

\(-oo(since, c-oo=-oo)\)

Hence,

underset \((x->5+)(lim)(1/x^(4/3)-1/(x-5)^4/3)=-oo\)

underset \((x->5^+)(lim)(1/x^(4/3)-1/(x-5)^(4/3))\)

It is known that

underset \((x->a)(lim)[f(x)+-g(x)]=underset(x->a)(lim)f(x)+-underset (x->a)(lim)g(x)\)

Then above can be written as

\(=underset(x->5+)(lim)(1/x^(4/3))-underset(x->5+)(lim)(1/(x-5)^4/3)\)

Plug Limits

\((1/5^4/3)-(oo)(since,(x-5)^(4/3)>0)\)

\(-oo(since, c-oo=-oo)\)

Hence,

underset \((x->5+)(lim)(1/x^(4/3)-1/(x-5)^4/3)=-oo\)