Show that lim_( x->a) (x^(3/8)-a^(3/8))/(x^(5/3)-a^(5/3))=9/40 a^(-31/24)?

2 Answers
Dec 5, 2017

See below.

Explanation:

lim_( x->a) (x^(3/8)-a^(3/8))/(x^(5/3)-a^(5/3))

(x^(3/8)-a^(3/8))/(x^(5/3)-a^(5/3))=a^(-31/24)((x/a)^(3/8)-1)/((x/a)^(5/3)-1)

now making a/x = y^24

a^(-31/24)((x/a)^(3/8)-1)/((x/a)^(5/3)-1) equiv a^(-31/24)(y^9-1)/(y^40-1)

but

(y^n-1)/(y-1) = 1+y+y^2+ cdots + y^(n-1) and then

a^(-31/24)(y^9-1)/(y^40-1) = a^(-31/24)(1+y+y^2+cdots+y^8)/(1+y+y^2+cdots+y^39)

and finally

lim_( x->a) (x^(3/8)-a^(3/8))/(x^(5/3)-a^(5/3))=lim_(y->1)a^(-31/24)(1+y+y^2+cdots+y^8)/(1+y+y^2+cdots+y^39) = 9/40a^(-31/24)

Dec 5, 2017

9/40*a^(-31/24).

Explanation:

We have, lim_(x to a) (x^n-a^n)/(x-a)=na^(n-1).

:."The Reqd. Lim. L="lim_(x to a)(x^(3/8)-a^(3/8))/(x^(5/3)-a^(5/3)),

=lim{(x^(3/8)-a^(3/8))/(x-a)}/{(x^(5/3)-a^(5/3))/(x-a)},

={3/8*a^(3/8-1)}/{5/3*a^(5/3-1)},

=3/8*3/5*a^(-5/8)/a^(2/3),

rArr L=9/40*a^(-31/24).