Find lim --> 0.5- (2x - 1)/(|2x^3 - x^2|)

Question

Find lim --> 0.5- (2x - 1)/(|2x^3 - x^2|)

Answer 1

lim_x→0.5[(2x-1)/ l2x³-x²l ]

 

= lim_x→0.5 [ (2x-1)/(x² l2x-1l) ]

 

    When x>0.5, 2x-1>0, so l 2x - 1 l = 2x - 1

    When x < 0.5, 2x-1<0, so l 2x - 1 l = -(2x - 1) 

 

Therefore, lim_x→0.5- [ (2x-1)/(x² l2x-1l ]

 

               = lim_x→0.5^- [(2x-1)/(x²(-(2x-1)))] = lim_x→0.5^- (-1/x²) = -4

 

    and         lim_x→0.5⁺ [(2x-1)/(x² l2x-1l)]

 

               = lim_x→0.5 ⁺ [(2x-1)/(x²(2x-1)) ] = lim_x→0.5 ⁺ (1/x²) = 4

 

Since the one-sided limits are not the same, the given limit does not exist.