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

Answer 1

Isaiah Allen

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)#

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Answer 2

Scarlett Allison

# 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).#

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Show that #lim_( x-&gt;a) (x^(3/8)-a^(3/8))/(x^(5/3)-a^(5/3))=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)#?