Drill: You will need to factor!

Drill: You will need to factor!

Drill answers

Lesson 2.2: Limits involving infinity Day 1 HW: p. 76: 1-20 Day 2 HW: P. 76: 21-40 P. 77: Quiz Quiz for AP Prep

Finite limits as x±∞ • As x∞ • As x gets larger and larger (look to the right side of the graph), what is the end behavior of a function? • As x-∞ • As x gets smaller and smaller(look to the left side of the graph), what is the end behavior of a function? • The line y = b is the horizontalasymptote of the graph of the function f(x) if either limx±∞ f(x) = b

Properties of Limits • Limx±∞ (1/x) = 0 • limx±∞ c = c, where c is a constant • Example: limx±∞ 2 = 2 • limx±∞ (c/xp) = 0, where c is a constant, and p is any whole number >1. • Example: limx±∞ (2/x2) = 0 • If the function f(x) is a rational expression where the degree in the denominator < degree in numerator, we say limx±∞ f(x) does not exist. • Example: limx±∞ (2x4 + 3x)/x2

Properties of Limits • If the function f(x) is a rational expression where the degree in the denominator > degree in numerator, we say limx±∞ f(x) is 0. • Example: limx±∞ (2x2 + 3x)/x3 = 0 • If the function f(x) is a rational expression where the degree in the denominator = degree in numerator, we divide leading coefficients to determine the limit • Example: limx±∞ (2x4 + 3x)/5x4 • limx±∞ f(x) is 2/5

Properties of Limitslimx±∞ f(x) = L and limx±∞ g(x) = M • Sum Rule • Difference Rule • Product Rule

Properties of Limits • Constant Multiple Rule • Quotient Rule • Power Rule

Examples • limx±∞ (2 + 1/x) • limx∞ (2 + 1/x) = limx∞ 2 + limx∞ 1/x = 2 + 0 = 2 • limx-∞ (2 + 1/x) = limx-∞ 2 + limx-∞ 1/x = 2 + 0 = 2 • Therefore, the horizontal asymptote is y = 2 • Use the graphs an tables to find limx±∞ • Graph: • Tables: There are 2 asymptotes: y = 1 and y = -1

Examples • Find limx∞ for f(x) = sinx/x • Use the sandwich rule • We know that -1 <sinx< 1. • So, for x > 0, • The limx∞ (-1/x) = 0 and limx∞ (1/x) = 0, so limx∞sinx/x = 0 • Find limx∞ (5x + sinx)/x • limx∞ (5x)/x + limx∞ (sinx)/x • 5 + 0 = 5

Definitions • Convergent: has a limit • Divergent: does NOT have a limit • Closure Identify the sequence as convergent or divergent. If it is convergent, state the limit. • 3, 15, 75, 375, . . . • 1, , , , , . . . • nth term = Divergent, r>1 Convergent to .5 Convergent to 7/8

Drill 1) Is the sequence defined by the formula convergent or divergent? If convergent, give its limit. 2) Determine the following: 3) Determine the following: Divergent, power in numerator > power in denominator Convergent to 0 Convergent to 6n/-3n = -2

Infinite limits as x a • If the values of a function outgrow all positive bounds as x approaches a finite number a, we say the limxa f(x) = ∞. • If the values become large and negative, we say limxa f(x) = -∞. • Referring to the graph below, we can see that as x 0- , the limit is ∞ and as x0+ , the limit is - ∞ • We say that the line x = 0 is the vertical asymptote,

Vertical Asymptote • The line x = a is a vertical asymptote of the graph of a function f(x) if either of the following is true. • limxa+ f(x) = ±∞ or • limxa- f(x) = ±∞ • Example: Find the vertical asymptotes of f(x) = 1/x2 Describe the end behavior to the left and to the right of each vertical asymptote. • The vertical asymptote is x = 0 • limx0+ f(x) = ∞ • limx0- f(x) = ∞

End Behavior Models • Modeling for |x| large • Let f(x) = 3x4 – 2x3 + 3x2 -5x + 6 and g(x) = 3x4 Show that while f and g are different for numerically small values of x, they are virtually identical for |x| large. • Graphically • Put both in calculator and repeatedly zoom out to show that as |x| gets larger, the graphs look virtually identical. • Analytically, (3x4 – 2x3 + 3x2 -5x + 6) / 3x4 • Limx±∞ (1 – 2/(3x) + 1/x2 – 5/(3x3 ) + 2/x4 ) • = 1 – 0 + 0 – 0 + 0 = 1

End Behavior Model • The function g is • A right end behavior model for f IFF limx∞ f(x)/g(x) = 1 • A left end behavior model for f IFF limx-∞ f(x)/g(x) = 1 • If one function provides both a left and right end behavior model, it is called an end behavior model. • Find an end behavior model for the following: • f(x) = (2x5 + x4 – x2 + 1)/(3x2 – 5x + 7) • Just use the leading terms in the numerator and denominator: (2x5)/(3x2) = (2/3)x3 . • Therefore, you could use (2/3)x3 to determine the end behavior of the given rational function.

Examples • Find an end behavior model for the following: • f(x) = (2x3 + x2 – x + 1)/(3x2 – 5x + 7) • Just use the leading terms in the numerator and denominator: (2x3)/(3x2) = (2/3)x . • Therefore, you could use (2/3)x to determine the end behavior of the given rational function.

Examples • Let f(x) = x + e-x . Show that g(x) = x is a right end behavior model for f while h(x) = e-x is a left end behavior model for f. (Will need to look at graphs!) • Right End Behavior Model: • limx∞ f(x)/g(x) = (x + e-x ) / x = • 1 + e-x / x = 1 + 0 = 1 • On the left • limx-∞ f(x)/h(x) = (x + e-x ) /e-x • = x/e-x + 1 = 0 + 1 = 1

Seeing limits as x±∞ • We can investigate the graph of f(x) as x  ±∞ by investigating the graph of y = f(1/x) as x0 • Find the limx∞ sin(1/x). • On the graph, it appears that limx∞ sin(1/x) = 0. • Now consider limx0+ sin(x). • The graph below indicates that limx0+ sin(x) = 0.

Convergent to 0 Convergent to -1.7 Divergent, grows without bound Convergent to 2 Convergent to 0 Divergent, grows without bound Convergent to -2.5 Divergent, grows without bound

Drill: You will need to factor!