Method 2: Dividing by the Highest Power of x

Option 2: Divide by the highest power of x.

The way you will most likely have to use on an in-class test.

Step 1: Determine if you are being asked to find a single limit, $\underset{x\to \mathrm{\infty }}{\lim }$ or $\underset{x\to –\mathrm{\infty }}{\lim }$, or if you are being asked to find the horizontal asymptotes of an equation, which means you have to apply both limits no matter what.

Step 2: Determine the single leading behavior (dominant term) over your entire equation. It may show up in multiple places in the equation.

Dominant term means for very, very large values of x , $\mathit{\pm }\mathit{\infty }$,which part of the equation will take over (dominate) the behavior of the equation. It may show up in multiple places in the equation.

This is quick list of the dominance from most dominant at the top to least dominant at the bottom:

1)      Exponential Equations: $\mathit{ }{5}^x, e^x$

2)      Powers of x:     $x^4, x^{1,000,000}$

3)      Logarithms: $\mathit{log}\left(x\right),\mathit{ ln}\left(x\right)$

It can sometimes be confusing when trying to compare terms outside of roots and inside of roots. Keep in mind $x^2=\sqrt{x^4}$. So, they are equal in terms of dominance.

Step 3: Divide every term in the equation by the dominant term, and simplify all the fractions you have just created.

If you have a root in your problem, what you divide by inside the root will be different than what you would divide by outside the root.

For example, if you had a square root in your problem, and $x^2$ is your dominant term you would divide everything outside the square root by $x^2$, and everything inside the square root by $x^4$.

Step 4: Apply a limit identity that you are given to each simplified term in your equation.

$\underset{x\to \infty }{\lim }\frac{1}{x}=0$

The limit identity is basically the bottom-heavy option from Step 3 of our Leading Behavior method.

While they give you this one version of the limit identity, it can be applied anytime you have a constant, k , divide by x to a positive power.

$\underset{x\to \infty }{\lim }\frac{1}{x}=\underset{x\to \infty }{\lim }\frac{7}{x^2}=\underset{x\to \infty }{\lim }\frac{113}{x^{10}}=\underset{x\to \infty }{\lim }\frac{k}{x^{+\mathit{power}}}=0$

Step 5: Draw your final conclusion about your limits based upon the Step 4 simplification, and give your final answer.