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Problem 43 Medium Difficulty

Find the limit, if it exists. If the limit does not exist, explain why.

$ \displaystyle \lim_{x \to 0.5^-}\frac{2x - 1}{| 2x^3 - x^2 |} $

Answer

(2x-1)/|2x^3-x^2|= (2x-1)/(x^2(|2x-1|) where X -> 0.5 minus, so
(2x-1) /|2x-1| = -1 therefore
= -1/x^2
? -4

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Video Transcript

to find the limit of two X -1 over the absolute value of two X cubed minus x squared. As X approaches 0.5 from the left, we first rewrite this into the limit as X approaches 0.5 from the left of two X -1 over. You have the absolute value of x squared times two x minus one. And we can read this further into the limit As X approaches 0.5 from the left of 2, -1 over we have absolute value of x squared Times Absolute Value of two X -1. Now, since expert is always positive, then it becomes limit as X approaches 0.5 from the left of 2, -1 over X squared times the absolute value of two x minus one. Now if X approaches 0.5 from the left then we're looking at values of X less than 0.5 or that's just two x less than one. If we multiply both sides by two and from here we get two X -1 Less than zero. And so The absolute value of two X -1, this is going to be equal to negative of two x minus one. Therefore we have Limit as X approaches 0.5 from the left of 2, -1 over x squared times negative of two x -1. And then from here we can cancel out the two X -1 and we're left with Limit as X approaches 0.5 from the left of negative one over X squared. And so evaluating this limit, we have negative one over 0.5 sq. This is equal to negative one over 0.25, or this is equal to -4. Therefore the limit exists and it's equal to negative four.