BCC.01.6 – Limits Involving Infinity

1. BCC.01.6 – Limits Involving Infinity MCB4U - Santowski

2. (A) Infinite Limits • Consider the function f(x) = x-2 and then the lim x0 (x-2) • If we try a direct substitution, we get (0)-2 which equals 1/02 which is undefined • So what is the limit??? • We can try a numeric approach and substitute in numbers close to but not equal to x = 0+ (like x = 0.001 and 0.000001) and x = 0- (like -0.00001 and -0.000001) • Then the values of f(x) are 1000, 1000000 for x = 0.001 and 0.000001 and 100000 and 1000000 for x = -0.00001 and -0.000001 • As it turns out, as x 0+ and 0-, the values of f(x) get larger and larger (f(x)  +∞) • So we do not reach a limiting number for f(x), meaning that this limit is undefined

3. (A) Infinite Limits – Graph of f(x) = x-2

4. (A) Infinite Limits - Summary • Consider which we said does not exist because the values of f(x) do not approach a number; i.e. the function does not reach a limiting value. • We are not regarding ∞ as a number, simply as a concept meaning "increasing without bound" or that the value of f(x) = x-2 can be made arbitrarily large as we get closer and closer to x = 0. • So we will write this as • And what we see on the graph is a vertical asymptote at x = 0

5. Work through these three examples numerically, graphically, and algebraically (B) Examples of Infinite Limits

6. (C) Limits at Infinity • In considering limits at infinity, we are being asked to make our x values infinitely large and thereby consider the “end behaviour” of a function • Consider the limit numerically, graphically and algebraically • We can generate a table of values and a graph (see next slide) • So here the function approaches a limiting value, as we make our x values sufficiently large  we see that f(x) approaches a limiting value of 1  in other words, a horizontal asymptote

7. Table of Values x y -10000.0000 1.00000 -6666.66667 1.00000 -3333.33333 1.00000 0.00000 -1.00000 3333.33333 1.00000 6666.66667 1.00000 10000.00000 1.00000 (C) Limits at Infinity – Graph & Table

8. Divide through by the highest power of x Simplify Substitute x = ∞  1/∞  0 (C) Limits at Infinity – Algebra

9. (D) Examples of Limits at Infinity • Work through the following examples graphically, numerically or algebraically • (i) • (ii) • (iii)

10. (E) Infinite Limits at Infinity • Again, recall that in considering limits at infinity, we are being asked to make our x values infinitely large and thereby consider the “end behaviour” of a function • Consider the limit lim x∞ ¼x3numerically, graphically and algebraically • We can generate a table of values and a graph (see next slide) • As it turns out, as x +∞ and as x -∞, the values of f(x) get larger and larger (f(x)  +∞) • So we do not reach a limiting number for f(x), meaning that this limit is undefined

11. A table of values: x y -1000 -250000000. -600 -54000000.0 -200 -2000000.00 200 2000000.000 600 54000000.00 1000 250000000.0 (F) Infinite Limits at Infinity – Graph and Table

12. (G) Examples of Infinite Limits at Infinity • Work through the following examples graphically, numerically or algebraically

13. (H) Internet Links • Limits Involving Infinity from Paul Dawkins at Lamar University • Limits Involving Infinity from Visual Calculus • Limits at Infinity and Infinite Limits from Pheng Kim Ving • Limits and Infinity from SOSMath

14. (I) Homework • Handouts from Stewart, 1997, Chap 2.9