Quick Sheet: Horizontal Asymptotes
1. In this worksheet, we examine how to compute
lim f (x)
x→±∞
If such a limit exists (call it L), then the line y = L is said to be a horizontal asymptote of f .
2. Unlike vertical asymptotes, it is possible for f to cross a horizontal asymptote.
3. A main template: Let r > 0. Then
lim
x→∞
1
=0
xr
(1)
• We have to be a little more careful for negative values of x. That is, as long as xr is defined, then
we get the same statement as above for x → −∞.
• The following is a useful fact for algebraic manipulation:
√
If x is negative, then x = − x2
4. Computation of limits at infinity:
• When given a quotient, divide numerator and denominator to take advantage of Equation 1.
• A difference of functions: “Rationalize” to make it a fraction, then use the previous idea.
• Templates:
y = tan−1 (x) has two horizontal asymptotes, y = π2 and y =
y = ex has y = 0 as a horizontal asymptote (for x → −∞).
−π
2
.
5. Some general rules to remember:
• If each item goes to infinity, so does its sum. The same CANNOT be said about a difference! Use
the idea given previously when taking the limit of a difference.
• If each item goes to infinity, so does the product. The same CANNOT be said about a quotient.
6. A general rule about the infinite limit of a rational function, p(x)/q(x).
• If the degree of p > degree of q, the limit is infinite.
• If the degrees are the same, the limit is the ratio of the leading terms.
• If the degree of p < degree of q, the limit is zero.
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