Question

Calculus: Evaluate lim x->0.5- (2x-1)/abs(2x^3 - x^2)

Answer 1

To clarify, the limit is: \[\lim_{x \rightarrow 0.5^-} \frac{2x-1}{|2x^3-x^2|}\]

Answer 2

Factor x^2 out of the denominator.
Write |A * B| as |A| * |B|. Then we will continue from there.

Answer 3

@aum I should mention that I got that far... but thanks. please help me deal with the |2x-1| part?

Answer 4

\[
\lim_{x \rightarrow 0.5^-} \frac{2x-1}{|2x^3-x^2|} = \lim_{x \rightarrow 0.5^-} \frac{2x-1}{|x^2(2x-1)|} = \lim_{x \rightarrow 0.5^-} \frac{2x-1}{|x^2| * |(2x-1)|}
\]
2x - 1 = 0 when x = 0.5
Slightly to the left of 0.5, 2x - 1 is negative.
When we approach 0.5 from the left (that is what \(x \rightarrow 0.5^-\) means), 2x -1 is negative and |2x-1| is positive. So the ratio is -1.

Answer 5

simplify the bottom and you'll see

Answer 6

\[
\lim_{x \rightarrow 0.5^-} \frac{2x-1}{|x^2| * |(2x-1)|} =
\lim_{x \rightarrow 0.5^-} \frac{1}{x^2} *
\lim_{x \rightarrow 0.5^-} \frac{2x-1}{(2x-1)|} = \frac{1}{0.5^2} * (-1) = -4
\]

Answer 7

you should have -1/x^2, so - 1/(1/2)^2 = -4

Answer 8

Thanks! I worked it out on my own, came back to this page, and you guys confirmed my answer.

Answer 9

Good. yw.