_ x ( x+8 x+1 )^ x+5 =

Step-by-step solution

To evaluate the limit _ x ( x+8 x+1 )^ x+5 , we identify that as x , the base x+8 x+1 1 and the exponent x+5 , producing the indeterminate form 1^ . For limits of this form, a standard approach involves rewriting in exponential form using the identity _ x a [f(x)]^ g(x) = e^ _ x a g(x)[f(x)-1] when the limit exists. Here, f(x) = x+8 x+1 and g(x) = x+5 . First compute f(x)-1 = x+8 x+1 - 1 = 7 x+1 . Then evaluate _ x (x+5) 7 x+1 = _ x 7x + 35 x+1 . Dividing numerator and denominator by x yields _ x 7 + 35 x 1 + 1 x =