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BCC.01.6 – Limits Involving Infinity MCB4U - Santowski (A) Infinite Limits • Consider the function f(x) = x-2 and then the lim x0 (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 (A) Infinite Limits – Graph of f(x) = x-2 (A) Infinite Limits - Summary 1 lim 2 • Consider x 0 x 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. 1 • So we will write this as lim x 0 x 2 • And what we see on the graph is a vertical asymptote at x=0 (B) Examples of Infinite Limits • 2 lim x 3 x 3 • lim log( x) x 0 lim tan( x ) • x 2 • Work through these three examples numerically, graphically, and algebraically (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 2 • Consider the limit lim x 1 numerically, graphically 2 and algebraically x x 1 • 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 (C) Limits at Infinity – Graph & Table • Table of Values • • • • • • • • x -10000.0000 -6666.66667 -3333.33333 0.00000 3333.33333 6666.66667 10000.00000 y 1.00000 1.00000 1.00000 -1.00000 1.00000 1.00000 1.00000 (C) Limits at Infinity – Algebra • x2 1 lim x x 2 1 x2 1 2 2 x lim x 2 x x 1 2 2 x x 1 1 2 x lim x 1 1 2 x 1 0 1 0 1 • Divide through by the highest power of x • Simplify • Substitute x = ∞ 1/∞ 0 (D) Examples of Limits at Infinity • Work through the following examples graphically, numerically or algebraically • (i) lim tan x 1 x 3x 2 x 2 • (ii) lim 2 x 5 x 4 x 1 • (iii) lim x x 2 x 2 (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∞ ¼x3 numerically, 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 (F) Infinite Limits at Infinity – Graph and Table • A table of values: • x • -1000 • -600 • -200 • 200 • 600 • 1000 y -250000000. -54000000.0 -2000000.00 2000000.000 54000000.00 250000000.0 (G) Examples of Infinite Limits at Infinity • Work through the following examples graphically, numerically or algebraically • • lim x x x 2 x2 x lim x 3 x (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 (I) Homework • Handouts from Stewart, 1997, Chap 2.9