# How do you find lim (t^3-6t^2+4)/(2t^4+t^3-5) as t->oo?

Jul 9, 2017

0

#### Explanation:

Finding the limit of a function is basically just a way to find out what value we get closer and closer to as we approach a certain number.

Finding the limit at infinity is no different. We should establish a couple of rules before we start.

$\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \overline{\underline{| \textcolor{w h i t e}{- - - -} \text{Rules} \textcolor{w h i t e}{- - - -} |}} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .$

$\textcolor{w h i t e}{- - - -} \cdot \frac{\infty}{n} = \infty \textcolor{w h i t e}{a a a a} \text{Where n is any integer}$

$\textcolor{w h i t e}{- - - -} \cdot \frac{n}{\infty} = 0$
$\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .$

Knowing this, we can go ahead and approach the problem as follows:

$\overline{\underline{\text{|Step 1|}}}$

The first thing when taking the limit of a rational function is that we should focus our attention on the denominator. There, we must look at the highest power of the polynomial, which in our case is color(red)[t^4.

• ${\lim}_{t \rightarrow \infty} \frac{{t}^{3} - 6 {t}^{2} + 4}{2 \textcolor{red}{\overline{\underline{|}}} {t}^{4} | + {t}^{3} - 5}$

$\overline{\underline{\text{|Step 2|}}}$

Next, we will take color(red)[t^4 and divide it by every term in both the numerator AND the denominator. Doing so we get the following:

• lim_(trarroo) ((t^3/color(red)[t^4]-"6t"^2/color(red)[t^4]+4/color(red)[t^4]))/((("2t"^4)/color(red)[t^4]+t^3/color(red)[t^4]-5/color(red)[t^4]))->lim_(trarroo) ((1/t-6/t^2+4/t^4))/((2+1/t-5/t^4))

$\overline{\underline{\text{|Step 3|}}}$

So, in our final step, we look at our rules that we noted above and simplify. Applying our rules, we get the following answer:

• ${\lim}_{t \rightarrow \infty} \left(\left(\stackrel{\textcolor{b l u e}{\text{0"cancel(1/t)-stackrelcolor(blue)"0"cancel(6/t^2)+stackrelcolor(blue)"0"cancel(4/t^4)))/((2+stackrelcolor(blue)"0"cancel(1/t)-stackrelcolor(blue)"0}}}{\cancel{\frac{5}{t} ^ 4}}\right)\right)$
$\textcolor{w h i t e}{a a a}$

• ${\lim}_{t \rightarrow \infty} \frac{\textcolor{b l u e}{0} - \textcolor{b l u e}{0} + \textcolor{b l u e}{0}}{2 + \textcolor{b l u e}{0} - \textcolor{b l u e}{0}}$
$\textcolor{w h i t e}{a a a}$

• ${\lim}_{t \rightarrow \infty} \frac{0}{2}$
$\textcolor{w h i t e}{a a a}$

• color(magenta)[ 0

"Answer":color(magenta)(0