Calc 2.5 - Limits involving infinity ex: lim x->0 1 x2 DNE BUT… we can also say: ex: lim x->0 1 x2 =  We are NOT regarding  as a # We are NOT saying the limit exists We ARE expressing the particular way in which the limit doesn’t exist…

or: f(x)-->  as x-->a General Notation lim f(x) =  “The limit of f of x as x approaches a is infinity”… x-->a or: f(x)-->  as x-->a “f of x approaches infinity as x approaches a” Negative  : lim - = - 1 x2 x-->a a x-->a- x-->a+ a One-Sided: lim f(x) =  x-->a+

Vertical Asymptotes “The line x = a is a vertical asymptote of the curve y = f(x) if y -->+ or - as x-->a(+/-) ex: Find lim x-->3+ 2 x - 3 and lim x-->3- Small +’s Small -’s x = 3 =  = -

Limits AT Infinity… x2 - 1 ex: lim x--> x2 + 1 1.1 -1 25 -1.1

OR lim f(x) = L x-->- Horizontal Asymptote “The line y = L is a horizontal asymptote of the curve y = f(x) if either: lim f(x) = L x--> OR lim f(x) = L x-->- lim f(x) = x-->-1- lim f(x) = x-->-1+ -1 2 1 lim f(x) = - x-->2- lim f(x) =  x-->2+ lim f(x) = 0 x-->- lim f(x) = 2 x-->

1 1 ex: Find lim x--> and lim x-->- x x General Law: “If n is a + integer then: lim x--> 1 xn = 0 AND lim x-->- Technique: Limit @ infinity of a rational function: Divide BOTH numerator AND denominator by the highest power of x in the DENOMINATOR… ex: lim x--> 3x2 - x - 2 5x2 + 4x + 1

[3x2 - x - 2] / x2 = 3 - 1/x - 2/x2 5 - 4/x + 1/x2 lim x--> ex: lim x--> Goes to 0 [5x2 + 4x + 1] / x2 = 3 5 x2 + 1 + x ex: lim x2 + 1 - x x--> = lim (x2 + 1) - x2 x--> x2 + 1 + x = lim 1 x--> x2 + 1 + x / x = 1/x 1 + 1/x2 + 1 lim x--> = 2

General Rule: lim ax = 0 for any a > 1 x -∞ lim 1/x = -∞ x0- = e-∞ as x 0- Evaluate lim e1/x x0- = 1 e ∞ = General Rule: lim ax = 0 for any a > 1 x -∞ Infinite Limits @ Infinity General notation: lim f(x) = ∞ x∞ “as x gets big, f(x) gets big” Other Examples: lim f(x) = -∞ x∞ lim f(x) = ∞ x-∞ lim f(x) = -∞ x-∞ “as x gets big, f(x) gets small” “as x gets small, f(x) gets big” “as x gets small, f(x) gets small”

HW 2.5 – pg 140 - #’s 1-3 all, 5, 13, 17-20 all, 31-33 all. ex: lim x3 = ∞ x∞ lim x3 = -∞ x-∞ lim ex = ∞ x∞ ≠ (∞ - ∞ ) ex: Find lim (x2 – x) x∞ = lim x (x – 1) x∞ = ∞ (∞ - 1) = ∞ = ∞ -1 ex: Find lim x∞ x2 + x 3 - x / x = x + 1 3/x - 1 lim x∞ = - ∞ HW 2.5 – pg 140 - #’s 1-3 all, 5, 13, 17-20 all, 31-33 all.

 Calc 2.5 !