AP Calculus AB : Asymptotic behavior in terms of limits involving infinity

Example Questions

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Example Question #1 : Asymptotic Behavior In Terms Of Limits Involving Infinity

Find all vertical asymptotes and horizontal asymptotes of the function,

The horizontal asymptote is,

The vertical asymptotes are,

Horizontal Aysmptotes

Vertical Asymptotes

Horizontal Aysmptotes

Vertical Asymptotes

Horizontal Aysmptotes

None

Vertical Asymptotes

The horizontal asymptote is,

The vertical asymptotes are,

The horizontal asymptote is,

The vertical asymptotes are,

Explanation:

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

Example Question #2 : Asymptotic Behavior In Terms Of Limits Involving Infinity

Undefined.

Explanation:

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.

Example Question #3 : Asymptotic Behavior In Terms Of Limits Involving Infinity

Undefined.

Explanation:

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.

Example Question #4 : Asymptotic Behavior In Terms Of Limits Involving Infinity

Undefined.

Explanation:

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.

Example Question #5 : Asymptotic Behavior In Terms Of Limits Involving Infinity

Undefined.

Explanation:

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.

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