Question

7-12 find the limit.

8.
`\lim _(t \rightarrow \infty) (t^(3)-t+2)/((2 t-1)\left(t^(2)+t+1\right))`

Answer 1

Divide through the limand by the highest power of t :

`\displaystyle \lim_(t\to\infty) (t^3-t+2)/((2t-1)(t^2+t+1)) = \lim_(t\to\infty) \frac{1 - \frac1{t^2} + \frac2{t^3}}{\left(2-\frac1t\right) \left(1+\frac1t + \frac1{t^2}\right)}`

(I distributed 1/t³ in the denominator as 1/t (2t - 1) and 1/t² (t² + t + 1).)

As to goes to infinity, these 1/tⁿ terms will converge to 0, and you're left with

`\displaystyle \lim_(t\to\infty) \frac{1 - \frac1{t^2} + \frac2{t^3}}{\left(2-\frac1t\right) \left(1+\frac1t + \frac1{t^2}\right)} = (1 - 0 + 0)/((2-0)(1+0+0)) = \boxed{\frac12}`