1) To find the horizontal asymptotes, find the limit of the function as , 

 Therefore, the function  has a horizontal asymptote 

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2) Vertical asympototes will occur at points where the function blows up, . For rational functions this behavior occurs when the denominator approaches zero. 

Factor the denominator and set to zero, 

So the graph of  has two vertical asymptotes, one at  and the other at .  They have been drawn into the graph of   below. The blue curves represent .

For this infinity limit, we need to consider the leading terms of both the numerator and the denominator.  In our problem, the leading term of the numerator is larger than the leading term of the denominator.  Therefore, it will be growing at a faster rate. 

Now, simply input the limit value, and interpret the results.

For infinity limits, we need only consider the leading term in both the numerator and the denominator.  Here, we have the case that the exponents are equal in the leading terms.  Therefore, the limit at infinity is simply the ratio of the coefficients of the leading terms. 

For infinity limits, we only consider the leading term in both the numerator and the denominator.  Then, we need to consider the exponents of the leading terms.  Here, the denominator has a higher degree than the numerator.  Therefore, we have a bottom-heavy fraction.  Even though we are evaluating the limit at negative infinity, this will still tend to zero because the denominator is growing at a faster rate.  You can convince yourself of this by plugging in larger and larger negative values.  You will just get a longer and smaller decimal.

Notice that as x increases, the points of data graphed appears to level out, flattening towards a certain value. This value is what is known as a horizontal asymptote. An asymptote is a value that a function approaches, but never actually reaches. Think of a horizontal asymptote as a limit of a function as x approaches infinity. In such a case, as x approaches infinity, any constants added or subtracted in the numerator and denominator become irrelevant. What matters is the power of x in the denominator and the numerator; if those are the same, then to coefficients define the asymptote. We see that the function flattens towards:

This matches the ratio of coefficients for the function:

Observation of the data points shows that there is a sharp increase and decrease on either side of a particular x-value. This type of behavior is observed when there is a vertical asymptote. An asymptote is a value that a function may approach, but will never actually attain. In the case of vertical asymptotes, this behavior occurs if the function approaches infinity for a given x-value, often when a zero value appears in a denominator. Noting this rule, the above function has a zero in the denominator at a definite point centered approximately around:

We find a zero denominator for the function:

Notice that as x increases, the points of data graphed appears to level out, flattening towards a certain value. This value is what is known as a horizontal asymptote. An asymptote is a value that a function approaches, but never actually reaches. Think of a horizontal asymptote as the limit of a function as x approaches infinity. In such a case, as x approaches infinity, any constants added or subtracted in the numerator and denominator become irrelevant. What matters is the power of x in the denominator and the numerator; if those are the same, then the coefficients define the asymptote. We see that the function flattens towards:

This matches the ratio of coefficients for the function:

Infinity limits can be found by only considering the leading term in both the numerator and the denominator. In this problem, the numerator has a higher exponent than the denominator.  Therefore, it will keep increasing and increasing at a much faster rate.  These limits always tend to infinity.

Firstly, recall the rules for evaluating the limits of rational expressions as they go to infinity:

If the degree on the top is greater than the degree on the bottom, the limit does not exist.

If the degree on the top is less than the degree on the bottom, the limit is 0.

If the degrees in both the numerator and denominator are equal, the limit is the ratio of the leading coefficients.

These rules follow from how the function grows as the inputs get larger, ignoring everything but the leading terms.

Then, note that, while the numerator is not written in the correct order for you, that the highest power in the numerator is 5. Likewise, the highest power in the denominator is 5. Thus, the limit will be the ratio of the coefficients on the  terms, which is .