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

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Answer 1 · 2017-12-05T09:30:06

See below.

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

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

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

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

Answer 2 · 2017-12-05T12:30:46

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

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

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

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

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

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

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

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