1. (4 points each) Compute each of the following limits or determine that the limit does not exist. If the limit does not exist but equals 0, say so. (a) lim (In(5x + x) ln(4(5*) + 7)). 81x (b) lim cos(x-5) x5 (x-5) (c) lim x1/x xx 2 - (x-5)(x-3). (e) lim (sin(x)) cos(x) x0+ (f) lim (ex +5x) 1/x xX (d) lim (cos(x))1/ x0+ (g) lim ln(1+2) x+0+ Warm-Up: Compute the limit below. lim xe2x+2x xxxex+1 } use 1.11 00 e2x + 2xe 2x+2 lim 2x worse than what we started with! = = lim Altemate: Trick: Divide ex + xex lim xex 2 + lim x + = + ezx ex by highest x+ ezx power of ex (ex) If do Recall, L'Hopital's Rule: = lim = 7 Zex =0 = or o indeterminate form, f+g diftble near a, 0 g'(x) 0 around a, then lim fix = lim f'(x) x-a g(x) x-a g'(x), Indeterminate Form: If lim flx) = lim 478 Strategy: Greu og(x)= x70 provided this new limit exists or equals. okay if DNE but One-sided limits exist ar are then lim fix)-g(x) has Rewrite - as a fraction & Oscillations BAD - indeterminate form. Only for subtraction then try to use ('Hopital (it possible) (common denom., rewrite function, log nus) Example. Compute each of the following limits or determine that the limit does not exist. If the limit does not exist but equals or -, say so. 1. lim lim 1 -x+1x 1-x +1+x - x 1 Form: x-x = lim x-it 1-| } Do NOT use 2.n. Form Cosy HIN = 2. lim (tan x- x/2- -sec x) lim Sinx Form: 00-00 = COSX COSX lim sinx1 = COS X } fom o L.H. = lim COSX = Sinx cos() -Sin (1) Q = o 0 2