Evaluate $\lim_{t \to \infty} \frac{(\sqrt t+ t^2)}{(4t - t^2)}$. Why is the limit as $t \to \infty = 0$? Would like some feedback on my process. Also, would like to gain a Calculus I understanding of limits. 
A question many are having a problem explaining to me is why is the limit as x approaches infinity designated as zero. Thats is, when I get to the point of solving a limit expression or function, why is it as x approaches infinity the value is designated zero?
Surely the number can be a very high positive number also? Or am I missing something here? for example why is the limit as x approached infinity $1/x = o$? What if x is 0.0001?  
 A: $$\lim_{t \to \infty} \dfrac{(\sqrt t+ t^2)}{(4t - t^2)}$$
$$=\lim_{t \to \infty} \dfrac{(1+t^{3/2})}{(4\sqrt t - t^{3/2})}$$
$$=\lim_{t \to \infty} \dfrac{(1+t^{3/2})}{\sqrt t(4 - t)}$$
$$=\lim_{t \to \infty} \dfrac{1}{\sqrt t(4 - t)} + \lim_{t \to \infty} \dfrac {t^{3/2}}{\sqrt t(4 - t)}$$
$$=0 + \lim_{t \to \infty} \dfrac {t}{(4 - t)}$$
$$=\lim_{T \to 0} \dfrac {1/T}{4-1/T},\; \text {with} \;T=1/t$$
$$=\lim_{T \to 0} \dfrac {1}{4T-1}$$
$$=-1$$
A: Dividing all terms by the highest power, $t^2$, is absolutely the right move.  Leaving you with:
$\cfrac{\cfrac{1}{t^{3/2}}+1}{\cfrac{4}{t}-1}$
Now evaluating this expression as t approaches infinity is easy, as long as you are okay with $\lim\limits_{x \to \infty} \cfrac{C}{x^n}=0$, C and n can be any positive real number!  If you'd like an explanation of this, I suggest asking a separate question.
$\cfrac{0+1}{0-1}=-1$ 
(You actually had the correct answer, but you got a bit lucky since you had errors along the way.
A: You have the right idea dividing through by the highest power, but something went a bit wrong after that... for one, what makes you so sure you can split limits up and take them separately over the numerator and denominator? You also had a mistake cancelling out your exponents - for example, the numerator in the upper-right corner should have reduced to $1/t^{3/2} + 1$. After that, in the denominator, you also have to distribute 1/t, not put it on only one of the two terms in the numerator.
