Limits Overview
Limit Notation (Left-Hand, Right-Hand, Overall)
Finding Limits Using a Graph
Finding a Limit Using Equations
Continuity

Example 4: Dividing by the Highest Power of x (Top-Heavy)

lim x x 4 + x 3 12 x 3 + 128 x 2

 

Step 1: Determine if you are being asked to find a single limit, lim x or lim x , or if you are being asked to find the horizontal asymptotes of an equation, which means you have to do both limits no matter what.

 

Here we are only being asked to find the single limit as x approaches infinity, x .

lim x x 4 + x 3 12 x 3 + 128 x 2

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

 

In this example x 4 is the dominant termof the entire equation because it is the highest power of x.

lim x x 4 + x 3 12 x 3 + 128 x 2

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

 

So here we are dividing every term by x 4 , and then simplifying those new fractions by canceling the x ’s.

lim x x 4 x 4 + x 3 x 4 12 x 3 x 4 + 128 x 2 x 4 = lim x 1 + 1 x 12 x + 128 x 2

 

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

lim x 1 x = 0

 

We will apply this limit identity to three different terms in the equation, 1 x , 12 x , and 128 x 2 . All three are heading to zero.

lim x 1 + 1 x 12 x + 128 x 2 = 1 + 0 0 + 0 = 1 0

 

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

 

Since the limit simplifies to 1 0 , this tells us the top of the fraction continues to grow and there is nothing the bottom can do to stop it. I like to think of this as 1 ( top-heavy ).

 

It is strange, but they definitely will accept this logical conclusion without any additional work.

lim x x 4 + x 3 12 x 3 + 128 x 2 = 1 0

lim x x 4 + x 3 12 x 3 + 128 x 2 =

Final Result:

lim x x 4 + x 3 12 x 3 + 128 x 2 =

Meaning:

The limit as x  approaches of x 4 + x 3 12 x 3 + 128 x 2 is y = .

 

There is no horizontal asymptote as x  approaches .

Remember you only know what is happening in that single direction, and cannot say anything about the other direction, .

 

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