Exam 31 - Degree Comparison for Rational Functions

ASSYMPTOTES. Degree Comparison for Rational Functions FOR HORIZONTAL: FOR VERTICAL: SET DENOM TO O; CONSTRUCTING A FUNCTION WI GIVEN Let f(x) = (p(x))/(q(x)) with degrees n and m: nm-limit=0 (horizontal asymptote: y = 0 ) nm limit leading coefficient ratio n>m limit = ±∞ (no HA; possible slant asymptote) ASYMPTOTES. x = a underline y = lim x e0 (- 1 + k/lambda) ^ (nx) = e ^ (k*mu) y = 4 x = - 1 Dominance Hierarchy e ^ x >> x ^ n >> ln(x) >> sin x, cos x x - a = 0 FIND THE LIMIT OF +00 x*xi*y INTERCEPTS (4x) CORE LIMIT5 Rules lim x -> 0 (sin x)/x = 1 Key Infinity Limits lim x -> ∞ 1/(x ^ b) = 0 lim x -> ∞ (ln(x))/x = 0 lim x -> 0 (tan x)/x = 1 lim x -> ∞ (sin x)/x = 0 lim x -> 0 (1 - cos x)/(x ^ 2) = 1/2 lim x -> ∞ arctan(x) = pi/2 TO SKETCH A GRAPH END BEHAVIOR BEHAVIOR AT ASSYMPTOTES lim x -> ∞ (1 + 1/x) ^ x = e y int x = 6 Xint y = 0 OLOCAL MIN UAX * (IFANY) LHOSPITAL 00 lim f(x) 0 = OR xa g(x) 00 lim x -> a (f' * (x))/(g' * (x)') lim x -> 0 (1 + i/x) ^ (x + 9) = e lim x -> ∞ (1 + k/x) ^ x = e ^ k EVAWATE BEFORE APPLYING. lim x -> 0 (sin(ax))/(sin(bx)) = a/b sinay ax sin bx + bx 0.00 1/00 2 L=lim x infty f (x) ^ 3 (x) L = c ln(L) = lim x -> 9 g(x) * ln(f(x)) ln f(x) g(x) * ln(f(x)) = (ln(f(x)))/(1/(g(x))) f(x) * g(x) = (f(x))/(1/(g(x))) = (g(x))/(1/(f(x))) 1F / (0 deg), 1 ^ (∞), ∞ * 0 deg WEIRD FACTORING (sin(x) - x)/(x ^ 3) = (x((sin x)/x - 1))/(x ^ y) MULTIPLY BY CONJUGATE 1o IF 6 ^ - OR -312 (3x ^ 2 + 5x)/(2x ^ 2 - 7x) = (x ^ 2 * (3 + 5 / x))/(x ^ 2 * (2 - 7 / x)) 7/60 = 0 0 ^ + Gin(fcx) =lim 1 1/f c x) factor OR FIND COMMON DENOM (sin 7x)/(sin 6x) = (sin 7x)/(7x) * (7x)/(6x) * (6x)/(5in * 6x) = 7/6 Power Rule n ne-1 integrate x ^ - 1 dx = ln|x| + C integrate x ^ n dx = (x ^ (n + 1))/(n + 1) + C_{1} Indefinite Integral integrate k dx = integrate k * x ^ 0 dx = kx + C x ^ x d/dx (x ^ x) = x ^ x * (ln(x) + 1) int x^ n dx= (x ^ (n + 1))/(n + 1) +C for n ne-1 integrate 1/x dx = ln|x| + C integrate e ^ x dx = e ^ x + C 2. . (2x) ^ x d/dx ((2x) ^ x) = (2x) ^ x * (ln(2x) + 1) d/dx (e ^ x) = e ^ x integrate cos x dx = sin x + C -3. arctan(ax) d/dx (arctan(ax)) = a/(1 + a ^ 2 * x ^ 2) integrate sin x dx = - cos x + C integrate sec^2 x dx = tan x + C integrate csc(x) * cot x dx = - csc(x) + C integrate sec(x) * tan x dx = sec(x) + C Differentiation Formula d/dx (k) = 0 Exponential Integrals Is integrate co * c ^ 2 * x dx = - cot x + C d/dx (x ^ n) = n * x ^ (n - 1) integrate e ^ (ax) dx = 1/a * e ^ (ax) + C integrate 1/(sqrt(1 - x ^ 2)) dx = arcsin(x) + C d/dx (ln|x|) = 1/x integrate a ^ z dx = (a ^ x)/(ln(a)) + C integrate 1/(1 + x ^ 2) dx = arctan(x) + C d/dx (sin x) = cos x int sin(ax) = - 1/a * cos(ax) + C integrate 1/(x * sqrt(x ^ 2 - 1)) dx = ae * c ^ - 1|x| + C d/dx (cos x) = - sin x int cos(ax) = 1/a * sin(ax) + C int sin(ax) = - 1/a * cos(ax) d/dx (tan x) = sec^2 x int sec^2 (ax) - 1/a * tan(ax) + C int cos(ax) = 1/a * sin(ax) d/dx (cncx) = - csc(x) * cot x int sec(ax) * tan(ax) = 1/a * sec(ax) + C int e ^ (ax) = 1/a * e ^ (ax) d/dx (sec(x)) = sec(x) * tan x int csc^2 (ax) = - 1/a * cot(ax) + C int a ^ x = (a ^ x)/(ln(a)) d/dx (cot x) = - csc^q x int csc(ax) * cot^psi (ax) - 1/a * csc(ax) + C int x ^ n = (x ^ (n + 1))/(n + 1) d/dx (arcsin(x)) = 1/(sqrt(1 - x ^ 2)) Inverse Trig Integrals int sec^2 (ax) = 1/a * tan(ax) d/dx (arctan(x)) = 1/(1 + x ^ 2) integrate 1/(1 + a ^ 2 * x ^ 2) dx = 1/a * arctan(ax) + C int 1/(1 + a ^ 2 * x ^ 2) = 1/a * arctan(ax) d/dx (arcsec|x|) = 1/(x * sqrt(x ^ 2 - 1)) integrate 1/(sqrt(1 - a ^ 2 * x ^ 2)) dx = 1/a * arcsin(ax) + C integrate 1/x = ln|x| dx