# Math Help - as time tends to infinity basic confusion!!!

1. ## as time tends to infinity basic confusion!!!

I dont think this answer bares any relevance to he situaton, M is mass t is time and k = 25root2

have condition M = kt/(1+t^2)^0.5

the answer as t tends to infinty is that M tends to k.

I have tried cancelling t's and this doesnt for me but i can only come to the conclusion that its alot bigger or smaller than that,, whats the relevance of 1 aswell.

2. The question is asking for $\lim_{t\to\infty}\frac{k t}{\sqrt{1+t^2}}$.

$\lim_{t\to\infty}\frac{k \sqrt{t^2}}{\sqrt{1+t^2}}$

$\lim_{t\to\infty}k\sqrt{\frac{t^2}{t^2+1}}$

$k\cdot\lim_{t\to\infty}\underbrace{\sqrt{\frac{t^2 }{t^2+1}}}_{\text{This limit is 1}}$

$k\cdot 1 = \boxed{~k~}$

There are lots of methods to find $\lim_{t\to\infty}\sqrt{\frac{t^2}{t^2+1}}$. You can use comparison of degrees and coefficients, L'hopital's rule, simplifying the limit etc.

Let $I = \lim_{t\to\infty}\sqrt{\frac{t^2}{t^2+1}}$.

$I\cdot I = \lim_{t\to\infty}\sqrt{\frac{t^2}{t^2+1}}\cdot\lim _{t\to\infty}\sqrt{\frac{t^2}{t^2+1}}$

$I^2 = \lim_{t\to\infty}\sqrt{\frac{t^2}{t^2+1}}\cdot\sqr t{\frac{t^2}{t^2+1}}$

$I^2 = \lim_{t\to\infty}\frac{t^2}{t^2+1}$

In $t^2 + 1$, $t^2$ will grow much faster than 1 and will make 1 meaningless. So we can cross it out:

$\lim_{t\to\infty}\frac{t^2}{t^2+\not 1} = \frac{t^2}{t^2} = 1$

So $I^2 = 1$ and $I=1$.