Prep Book Chapter 3 - Limits of Functions Ex: Investigate the behavior of function f defined by: f(x) = (x2 - x + 2) for values of x near 2… The closer to 2 x is, the closer to 4 f(x) is. So we write: And say: “The limit of f(x) as x approaches a = L” General notation:

The limit of f(x) as x approaches a = L IF we can get f(x) as close to L as we want by taking x as close to a as we want, BUT NOT EQUAL TO a (x  a) Alternate notation for : ex: Guess the value of: 1 x L .5

One-Sided Limits a a “Left-Hand Limit” “Right-Hand Limit” x x x x x x or “As x approaches a from the left…” “As x approaches a from the right…” a a “Left-Hand Limit” “Right-Hand Limit” x x x x x x

The left and right hand limits must be equal for the overall (2-sided) limit to exist…

ex: Find if it exists. Does Not exist (DNE)

IF c is constant and lim f(x) and lim g(x) exist then… 1. lim [f(x) + g(x)] = lim f(x) + lim g(x) x->a “The limit of a sum (or diff) is the sum (or diff) of the limits” “The limit of a constant times a function is the constant times the limit of the function.” 2. lim [cf(x)] = c lim f(x) x->a x->a 3. lim [f(x)g(x)] = lim f(x) • lim g(x) “The limit of a product is the product of the limits” (= 0) lim f(x) lim g(x) x->a f(x) g(x) 4. lim = “The limit of a quotient is the quotient of the limits”

ex: Use the limit laws and the graphs of f and g below to evaluate the following limits if they exist… a) lim [f(x) + 5g(x)] x->-2 f g b) lim [f(x)g(x)] x->1 c) lim x->2 f(x) g(x)

Additional Laws lim [f(x)] = lim f(x) n 5. Power Law: x->a n 5. Power Law: 6. Special Limits: lim x = a x->a n lim x = a x->a n lim c = c x->a lim x = a x->a 7. Root Law: lim f(x) = lim f(x) x->a n (where n is a positive integer)

“If f is a polynomial or rational function and a is in the domain of f then”:… lim f(x) = f(a) x->a Direct Substitution ex: lim 2x2 - 3x + 4 = [2(5)2 - 3(5) + 4] = 39 x->5 “Continuous at a” (x + 1)(x - 1) x - 1 lim x->1 lim x->1 x2 - 1 x - 1 = lim (x + 1) = 2 x->1 =

(3 + h)2 = (3 + h)(3 + h) = [9 + 6h + h2] ex: Evaluate lim h-->0 (3 + h)2 - 9 h - 9 (3 + h)2 = (3 + h)(3 + h) = [9 + 6h + h2] 6h + h2 h = lim h-->0 = h (6 + h) h lim h-->0 = lim h-->0 (6 + h) = 6 t2 + 9 ex: Find lim t-->0 - 3 t2 t2 + 9 + 3 = (t2 + 9) - 9 t2 t2 + 9 + 3 t-->0 lim = t2 t-->0 lim t2 + 9 + 3 = t-->0 lim 1 t2 + 9 + 3 1 = t-->0 lim 9 + 3 6

If f(x) < g(x) when x is near a (except possibly at a), and the limits of f and g both exist as x approaches a, then: lim f(x) < lim g(x) x-->a x-->a The Squeeze Theorem If f(x) < g(x) < h(x) when x is near a (except possibly at a) and: lim f(x) = lim h(x) = L x-->a x-->a then… lim g(x) = L x-->a

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… 13

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+ 14

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 =  = - 15

Limits AT Infinity… x2 - 1 ex: lim x--> x2 + 1 1.1 -1 25 -1.1 16

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”

∞ 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∞ = - ∞

Trig Identities and Special Limits Additional Rules: Pythagorean Identity: