lim t-> -infinity for sqrt(4t^2-5t-1)/(2t-3)
I get infinity or -infinity, but the real answer is -1. How does this occur?
as x gets large, higher powers dominate, so
√(4t^2-5t-1) -> √(4t^2) = |2t|
(2t-3) -> 2t
make sense?
To find the limit of the given expression, we can divide both the numerator and the denominator by the highest power of t in the expression, which is t^2.
Therefore, we rewrite the expression as follows:
lim t-> -∞ (sqrt((4t^2-5t-1)/(t^2))/(2t/t^2-3/t^2))
Simplifying further, we get:
lim t-> -∞ (sqrt(4-(5/t)-(1/t^2))/(2/t-3/t^2))
Now, as t approaches -∞, both (1/t^2) and (5/t) tend to zero. Therefore, the expression becomes:
sqrt(4-0-0)/(0-0) = sqrt(4)/0 = ∞
Hence, it seems like the limit should be infinity. However, it is important to note that taking a square root can introduce negative values. In this case, the expression inside the square root, (4t^2-5t-1), becomes negative for some values of t, specifically when t < -1/2.
For t < -1/2, the expression inside the square root becomes negative, leading to an imaginary value in the numerator. Since the denominator approaches zero as t approaches -∞, we cannot use L'Hôpital's rule to further simplify the expression.
Therefore, in this case, the limit is not defined, and we cannot conclude that it is either infinity or -infinity.