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

Example 1: Leading Behavior (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 leading behavior (dominant term) on the top and bottom of your fraction.

 

In this example x 4 is the dominant termon the top of the fraction because it is the highest power of x. The dominant term on the bottom of the fraction is x 3 because it is the highest power of x. The 12 comes along for the ride because it is a part of the x 3 term, but it won’t factor in to our final decision.

x 4 12 x 3

Step 3: Determine which of the three possible behavior scenarios you are working with.

 

Since the x 4 is the larger power of x, x 4 is the dominant term. This dominant term is only on the top of the fraction you would call this a Top-Heavy situation.

x 4 12 x 3 = Top Heavy

Step 4: Draw your final conclusion about your limits based upon the Step 3 scenario, and give your final answer.

 

Always be careful with the final sign of your . Here the power of the dominant term is even , which would make it positive, and the term, x 4 , itself is positive (i.e., not x 4 ).

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

 

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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