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Show that lim x->a (x^3/8-a^3/8)/(x^5/3-a^5/3)?

Answer 1

Emma Aldridge

#lim _(x->a) (x^3/8-a^3/8)/(x^5/3-a^5/3)=(9) /(40a^(2)) #

#lim _(x->a) (x^3/8-a^3/8)/(x^5/3-a^5/3)#

As we can easily recognize that this is #0/0# we'll modify the fraction

#((x^3-a^3)*3)/((x^5-a^5 )*8)#

Apply the factoring rule

#(cancel(x -a)(a^2+ax+x^2)*3)/(8cancel(x-a)(x^4+x^3a+ x^2a^2+xa^3+a^4)#

Plug in the value a

#((a^2+aa+a^2)*3)/(8(a^4+a^3a+ a^2a^2+aa^3+a^4)#

#((3a^2)*3)/(8(2a^4+2a^3a^1+ a^2a^2)#

#(9a^2) /(8(2a^4+2a^4+ a^4)#

#(9a^2) /(8(5a^4)#

#(9a^2) /(40a^4)#

#= (9) /(40a^(4-2))#

#= (9) /(40a^(2))#

#lim _(x->a) (x^3/8-a^3/8)/(x^5/3-a^5/3)=(9) /(40a^(2)) #

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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)?

Show that lim x->a (x^3/8-a^3/8)/(x^5/3-a^5/3)?