Method 1: Leading Behavior

Option 1: Leading behavior

The quickest and easiest way.

Great for using on the multiple-choice section of the AP Calc Test and for checking your answer if you are forced to use Option 2.

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 do both limits no matter what.

The result of all of these will be a horizontal asymptote, but with a single limit you only know the horizontal asymptote in that specific direction.

Step 2: Determine the leading behavior (dominant term) on the top and bottom of your fraction.

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.

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 $x^2=\sqrt{x^4}$.

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

1)      Top-Heavy : The dominant term is located only on the top of the fraction. In this situation the top is growing and the bottom cannot do anything to stop it. The answer to the limit in this situation would always be $y=\mathit{\pm }\mathit{\infty }$.

This means that there is NO horizontal asymptote.

2)      Bottom-Heavy : The dominant term is located only on the bottom of the fraction. In this situation the bottom of the fraction is growing and the top cannot do anything to stop it. The answer to the limit in this situation would always be $y=0$.

3)      Balanced : The dominant term is on the top and the bottom of the fraction. In this situation the top and the bottom of the fraction is growing at the same rate. The answer to the limit in this situation would always be the coefficients (numbers in front) of those dominant terms, $y=\frac{a}{b}$.

Remember as you do this to keep track of what the final sign would be. Negatives (i.e., $\mathit{–}\mathit{\infty }$) taken to an even power become positive, but taken to an odd power stay negative.

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