Quick Sheet Horizontal Asymptotes | Cheat Sheet

University:
Whitman College
Course:
MATH-125 | Calculus I
Academic year:

2020
Views:

341

Pages:

1
Author:

LiaistdiCemr

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. 1

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