Question #c1d5e

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Answer 1 · 2016-10-24T19:05:05

The limit does not exist

The limit #lim_(x->a)f(x)# exists if and only if the right and left hand limits exist and are equal, that is, if and only if

#lim_(x->a^+)f(x) = lim_(x->a^-)f(x)#

Note that

#|2x^3-x^2| = |x^2(2x-1)|#

#= x^2|2x-1|#

#= {(-x^2(2x-1) if x <= 1/2),(x^2(2x-1) if x >= 1/2):}#

With that, let's calculate the two one-sided limits at #1/2#.

As we approach #1/2# from the left:

#lim_(x->1/2""^-)(2x-1)/|2x^3-x^2| = lim_(x->1/2""^-)(2x-1)/(-x^2(2x-1))#

#=lim_(x->1/2""^-)-1/x^2#

#=-1/(1/2)^2#

#=-4#

As we approach #1/2# from the right:

#lim_(x->1/2""^+)(2x-1)/|2x^3-x^2| = lim_(x->1/2""^+)(2x-1)/(x^2(2x-1))#

#=lim_(x->1/2""^+)1/x^2#

#=1/(1/2)^2#

#=4#

As #4!=-4#, the left and right hand limits are not equal, meaning the limit does not exist.