Find the limit.
lim
t → ∞
(sqrt t + t^2)/(6t − t^2)
when t --oo
top --> t^2
bottom --> -t^2
result --> -1
try it with some big numbers :)
Thank you! That was a huge help!
To find the limit as t approaches infinity of (sqrt t + t^2)/(6t - t^2), we can use algebraic manipulation.
Step 1: Simplify the expression
(sqrt t + t^2)/(6t - t^2) = (t^(1/2) + t^2)/(6t - t^2)
Step 2: Divide both the numerator and denominator by t^2, which is the highest power of t
(t^(1/2)/t^2 + t^2/t^2) / ((6t/t^2) - (t^2/t^2))
This simplifies to:
(t^(-3/2) + 1)/(6/t - 1)
Step 3: Take the limit as t approaches infinity
Now that we have simplified the expression, we can find the limit as t approaches infinity.
(t^(-3/2) + 1)/(6/t - 1)
Since the numerator and denominator both have t terms, as t approaches infinity, the t term dominates. So we can ignore the other terms.
As t becomes very large, t^(-3/2) approaches 0 and 1/(6/t - 1) becomes 1.
Therefore, the limit, as t approaches infinity, is 0.