Chapter 2. Limits and Continuity 2.6 Limits Involving Infinity; Asymptotes of Graphs

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1 2.6 Limits Involving Infinity; Asymptotes of Graphs Chapter 2. Limits and Continuity 2.6 Limits Involving Infinity; Asymptotes of Graphs Definition. Formal Definition of Limits at Infinity.. We say that f(x) has the it L as x approaches infinity and we write x + if, for every number ɛ > 0, there exists a corresponding number M such that for all x x > M f(x) L < ɛ. 2. We say that f(x) has the it L as x approaches negative infinity and we write x if, for every number ɛ > 0, there exists a corresponding number N such that for all x x < N f(x) L < ɛ.

2 2.6 Limits Involving Infinity; Asymptotes of Graphs 2 Definition. Informal Definition of Limits Involving Infinity.. We say that f(x) has the it L as x approaches infinity and write x + if, as x moves increasingly far from the origin in the positive direction, f(x) gets arbitrarily close to L. 2. We say that f(x) has the it L as x approaches negative infinity and write x if, as x moves increasingly far from the origin in the negative direction, f(x) gets arbitrarily close to L. Example. Example page 04. Show that x x = 0. Solution. Let ɛ > 0 be given. We must find a number M such that for all x > M x 0 = x < ɛ. The implication will hold if M = /ɛ or any larger positive number (see the figure below). This proves x = 0. We can similarly prove that x

3 2.6 Limits Involving Infinity; Asymptotes of Graphs 3 x x = 0. QED Figure 2.32, page 02 of the th Edition Theorem 2. Rules for Limits as x ±. If L, M, and k are real numbers and and g(x) = M, then. Sum Rule: (f(x) + g(x)) = L + M 2. Difference Rule: (f(x) g(x)) = L M 3. Product Rule: (f(x) g(x)) = L M 4. Constant Multiple Rule: (k f(x)) = k L

4 2.6 Limits Involving Infinity; Asymptotes of Graphs 4 f(x) 5. Quotient Rule: g(x) = L M, M 0 6. Power Rule: If n is a positive integer, then (f(x))n = L n. n x c 7. Root Rule: If n is a positive integer, then n is even, we also require that x c f(x) = L > 0). f(x) = n L = L /n (if Example. Page 4 number 4 and Page 5 number 36. Definition. Horizontal Asymptote. A line y = b is a horizontal asymptote of the graph of a function y = f(x) if either f(x) = b or f(x) = b. x x Example. Page 5 number 68, find the horizontal asymptotes. Example. Page 06 Example 5. Prove x ex = 0.

5 2.6 Limits Involving Infinity; Asymptotes of Graphs 5 Definition. Oblique Asymptotes. If the degree of the numerator of a rational function is one greater than the degree of the denominator, the graph has an oblique asymptote (or slant asymptote). The asymptote is found by dividing the denominator into the numerator to express the function as a linear function plus a remainder that goes to zero as x ±. Example. Page 6 number 02, find the slant asymptote. Example. Page 6 number 86. Definition. Infinity, Negative Infinity as Limits. We say that f(x) approaches infinity as x approaches x 0, and we write f(x) =, x x 0 if for every positive real number B there exists a corresponding δ > 0 such that for all x 0 < x x 0 < δ f(x) > B. 2. We say that f(x) approaches negative infinity as x approaches x 0,

2.6 Limits Involving Infinity; Asymptotes of Graphs 6 and we write f(x) =, x x 0 if for every negative real number B there exists a corresponding δ > 0 such that for all x 0 < x x 0 < δ f(x) < B.

6 2.6 Limits Involving Infinity; Asymptotes of Graphs 6 and we write f(x) =, x x 0 if for every negative real number B there exists a corresponding δ > 0 such that for all x 0 < x x 0 < δ f(x) < B. Figures 2.40 and 2.4, 0th Edition. Note. Informally, x x0 f(x) = if f(x) can be made arbitrarily large by making x sufficiently close to x 0 (and similarly for f approaching negative infinity). We can also define one-sided infinite its in an analogous manner (see page 6 number 93).

7 2.6 Limits Involving Infinity; Asymptotes of Graphs 7 Definition. Vertical Asymptotes. A line x = a is a vertical asymptote of the graph if either x a f(x) = ± or f(x) = ±. + x a Note. Recall that we look for the vertical asymptotes of a rational function where the denominator is zero (though just because the denominator has zero at a point, the function does not necessarily have a vertical asymptote at that point). We make things more precise in the following result: Dr. Bob s Infinite Limits Theorem. Let f(x) = p(x) q(x). Suppose p(x) = L 0, x x 0 x x0 q(x) = 0, and q(x) is of the same sign in some open interval containing x 0. Then x x0 f(x) = ±. We can say something similar for one-sided its. Note. We can simplify Dr. Bob s Infinite Limits Theorem by applying it to rational functions. It then becomes: Let f(x) = p(x) be a rational function. Suppose p(x) = L 0 and q(x) = 0. Then q(x) x x + 0 x x + 0 f(x) = ±. We can say something similar for its from the left x x + 0 and for two-sided its. Examples. Page 5 numbers 54, Page 6 number 02 (again), Page 5 number 74, Page 6 number 96, and Page 2 Example 8.

LIMITS AT INFINITY MR. VELAZQUEZ AP CALCULUS

LIMITS AT INFINITY MR. VELAZQUEZ AP CALCULUS RECALL: VERTICAL ASYMPTOTES Remember that for a rational function, vertical asymptotes occur at values of x = a which have infinite its (either positive or