lim (x->0.5 from the left)((2x-1)/|(2x^3-x^2)| = -4
I would have thought positive four. How can this be?
Look at the denominator. For values of x>1/2, the x^3 term donminates. However, for values less than 1/2, the x^2 term donminates and will force the denominator negative.
I see that now. Thank you.
To find the limit of the expression (2x - 1) / |(2x^3 - x^2)| as x approaches 0.5 from the left, we need to evaluate the expression as x approaches 0.5 but stays slightly to the left side of 0.5.
The expression (2x - 1) / |(2x^3 - x^2)| can be simplified as follows:
When x is approaching 0.5 from the left, the numerator (2x - 1) is approaching (2(0.5) - 1) = 0.
To simplify the denominator, we factor it:
2x^3 - x^2 = x^2(2x - 1)
As x approaches 0.5 from the left, x^2 approaches (0.5)^2 = 0.25.
Similarly, (2x - 1) approaches [(2)(0.5) - 1] = 0.
So, the denominator (2x^3 - x^2) approaches (0.25)(0) = 0.
Now, when we divide 0 by 0, the result is undefined. Therefore, we have an indeterminate form when evaluating the limit.
Hence, the limit (x->0.5 from the left)((2x - 1) / |(2x^3 - x^2)| is not equal to either positive or negative four because it is undefined in this case.