Question #c1d5e

1 Answer
Oct 24, 2016

The limit does not exist

Explanation:

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.