L’Hopital’s Rule for Indeterminate Forms - Homework Basic Problems - calculate your answers and check on your calculators 2x2 " x " 3 x !"1 x !1 1. 4 x "1 " "5 lim x !"1 1 lim x2 " x " 6 x !3 x"3 2. 2x "1 "5 lim x !3 1 lim lim x2 " x " 3 "3 3. x !3 x " 2 x !1 lim 4. lim x !1 x "1 x2 ! 3 " 2 1 2x 2 x2 ! 3 "2 x !1 2x " lim 2 x2 ! 3 2 " r2 ! 3 5. "3r 2 " lim 3r r 2 ! 3 " 6 lim r !1 r !1 2r " 2 r2 ! 3 x 3 " 3x ! 2 lim 3 x !1 x " x 2 " x ! 1 3x 2 " 3 6. lim x !1 3 x 2 " 2 x " 1 6x 3 lim " x !1 6 x " 2 2 1 " x ! ln x x 3 " 3x ! 2 7. "1 x 2 "1 "1 ! 1 x " lim " lim 2 x !1 3 x " 3 x !1 6 x 6 x 1 " ex 8. 1 lim x " "1 x !0 "e lim 1" r 3 r !1 lim x !1 © www.MasterMathMentor.com BC Solutions lim x !0 - 12 - Illegal to post on Internet x cos x ! sin x x WE!! x !! sin x " $ cos x ! cos x #0 lim x (0 1 sin 2 x tan x 3x -XE!! sin 2 x sec 2 x $ 2 tan x cos 2 x lim #0 x (0 3 sin 2 x + tan x 6x --E!! 2 cos 2 x $ sec 2 x 1 lim # x (0 6 2 tan x ! x x ! sin x sec 2 x ! 1 lim x (0 1 ! cos x -GE!! 2 sec 2 x tan x lim x (0 sin x 2 sec 2 x sec 2 x $ 4 sec 2 x tan 2 x lim #2 x (0 cos x e x ! e! x ! 2 x x (0 x ! sin x x e $ e! x ! 2 lim x (0 1 ! cos x -IE!! e x ! e! x lim x (0 sin x e x $ e! x #2 lim x (0 cos x 10 2 x ! 2 x ! 10!2 x lim x (0 10 2 x ! 10!2 x 2 ln10 5 10 2 x ! 2 $ 2 ln10 5 10!2 x -JE!! lim x (0 2 ln10 5 10 2 x $ 2 ln10 5 10!2 x 4 ln10 ! 2 2 ln10 ! 1 # 4 ln10 2 ln10 lim x (0 lim x (0 lim x (0 lim x (0 lim x2 !1 1 # -KE! lim 2 x (4 4 x $ x 4 2x2 $ 4 x ! 7 #0 -LE!! x (4 x 3 $ 3 x 2 ! 5 lim !"###$%&'()*%&(+%),(-*$.-/!!""#!$%&'()%*+!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!,!-I!,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!""#$%"! (%! .%+(! %*! /*(01*0( x3 3x 2 6x 6 -TE!! lim x # lim x # lim x # lim x # 0 x (4 e x (4 e x (4 e x (4 e x2 !1 1 # !!Md%(0>!YiQ%.)(6&!?%*i(!?%1FO -UE!! lim x (4 2 x $ 1 2 x2 !1 1 x2 # lim # !4 -WE!! lim x (4 2 x $ 1 2 x (!4 x $ 1 1 ln x # lim x # lim! x # 0 GXE!! lim x (0 1 x (0 !1 x (0 2 x x ex ex # lim # lim xe x # 4 G-E!! lim x (4 ln x x (4 1 x x (4 x2 lim x 2e!3 x # lim 3 x x (4 x (4 e GIE!! 2x 2 lim 3 x # lim 3 x # 0 x (4 3e x (4 9e lim ! x 2 ln x " x (0 $ GKE! lim$ x (0 ln x 1x # lim$ 2 3 x (0 !1 x 1x lim !! x 2 " # 0 lim x GGE!! x (4 e lim x( GJE!! 6 2 lim x( 6 2 x 1 #0 x (4 2 xe x # lim cos2 3 x sec x lim # sec 2 3 x x ( 6 cos x 2 !6 cos 3 x !! sin 3 x " #0 ! sin x .1 ! cos x 1 lim!csc x ! cot x " # lim0 3 x (0 x (0/ sin x 2 GLE!! . sin x 1 lim0 3#0 x (0/ cos x 2 x (0 $ !"###$%&'()*%&(+%),(-*$.-/!!""#!$%&'()%*+!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!,!-J!,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!""#$%"! (%! .%+(! %*! /*(01*0( 7 1 7 x ! sin x : 1: lim9 ! < # lim9 x (08 sin x x ; x (08 x sin x <; 7 1 ! cos x : GTE!! lim9 x (08 x cos x $ sin x < ; 7 : ! sin x lim9 <#0 x (0 x !! sin x " $ cos x $ cos x 8 ; 7 x ! 1 ! ln x : 7 1 1 : # lim9 lim9 ! < < x (18 ln x x ! 1; x (18 ! x ! 1" ln x ; : 7 9 1!1 x < 7 x !1 : < # lim9 9 < GUE!! lim x (1 ! x ! 1" < x (18 x ! 1 $ ln x ; 9 ln $ x <; 98 x 7 1 : 1 lim9 <# x (1 1 $ 1 x 8 ; 2 V1*#, %;W%.J#;, AB?1@2)%"B/, =<"#, F*15"#(/, 6, 7%"J<"%)#, '1<*, %./8#*/, %.;, JM#J9, 1., '1<*, J%"J<"%)1*/ lim x !x 2 " x (0 *y#x !x 2 " lim!1 $ x " * y # !1 $ x " 1 x ln!1 $ x " x IXE!! ln!1 $ x " 1 # lim #0 lim x (4 x (4 x 1$ x ln y # 0 * y # e ln y # Technically, this is a right - sided limit sin x x x (0 ln y # x 2 ln x ln x 1x # lim #0 GWE!! lim x (0 1 x 2 x (0 !1 x 3 ln y # 0 * y # 1 .11 lim$ 0 3 x (0 / x 2 1 sin x .11 * y #0 3 / x2 .11 ln y # sin x ln0 3 # ! sin x ln x / x2 ! ln x !1 x lim # lim I-E!! x (0 $ 1 sin x x (0 $ ! cos x sin 2 x lim !tan x " x(6 cos x * y # !tan x " cos x 2! ln y # cos x lntan x sec 2 x tan x lntan x # lim IGE!! x ( 6 ! 1 cos x x ( 6 ! sin x cos2 x 2 2 lim sin 2 x 2 sin x cos x lim$ # lim$ #0 x (0 x cos x x (0 x !! sin x " $ cos x cos x sec 2 x cos2 x 5 # lim #0 6 6 x( ! sin tan x x ( 2 ! sin 2 x 2 ln y # 0 * y # 1 ln y # 0 * y # 1 lim X ! Y):)(+!%3!(80!(9.0!! A!! X A!!X!A!!!g!!A!!!C!!!610!.1)!)*70(01:)*6(0!3%1:+E!!H)*7!(80!3%&&%?)*<!B9!)*+.04()%*> IIE!! lim$ x (0 x #0 ln x IIE!! lim ! !cos x " x (! 6 " 2 tan x # 0 !!!! . 2 1 $ tan x 3 # 4 IJE!! lim ! 0 2 x (! 6 2" / 6 ! 2 x !"###$%&'()*%&(+%),(-*$.-/!!""#!$%&'()%*+!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!,!-K!,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!""#$%"! (%! .%+(! %*! /*(01*0(