10-29-21
14.2 - Limits_ond Confinuity in several variables
Let f(xy) be a funcfion of two variables and (a,b) be a foint n theplane.We writeumf(xy)=if f(xy) isasclose os x=ab we want to L,whenover (ay) is sufficienHy cloxe to (a,b) xnfoursly: for any >O taere 1s d0 such that distance (y2,(a,b<&=[o)-4k
j Determine if the lim f(an exists and if it does calculate it xya5 fif umf(xy)=L,then f(xy)-L as (aiy)-(a,b) along any curve C q0(x
ab=0,01
C-aloy x-axis
approaches
if thwre are fwo PaHns C and C such that f(xy)=L,as (ay)-(ab) a\ong C and f(x)-L as (xy)-(ab) alon Cz and the twO Limifs are diff (L,+L),then Lim f(xy) does not exis+ (DNE) x1y=(ab) Exl Show that lim 2+32 DNE (xy>00 fxy=y Limit along the x axis um alony y axis =(oxj 20 O=(ox)jn) X-70 Ys#O yo 13
Ex2
Show that Lim (x160 fx=xy f(x=o=fy Limf(rc=Olimf(0y) y-o doesnt prove Hhaf lim ONE so hove to use new Line
y=mx Um alcng y=mx f(xmx=x8m)mx xmxx(i+m) Lim f(xmx)-mdepenals on ge X>O 1+mmifi+will bezero fm=1weset Lmfxx #O SoLim DNE
Alternafive Soluhch to Ex 2 -use polar coorclinafes S x=rcose Thenx-(o0=7r>0 Ly=rsine f(rcose,rsine) = (rcase)(rsine) : cosesino (rco(rsino Lim f(rose,rsine) = cosesine defends on angle e ro For-0 Fore=Yq Lim=O im== O#Yz SO &im DNE
Ex3 Shoco that Lm XayT DNG xy00 =+x Lim alory y=mx
Lmf(xmx)=m0=O xC O+m aif x=yz Hhen f(xy)=Vz takex=+y=t (+hen x3=+=y and(xy)-0asf-O PRRRTITOIE dim along the Path (+,t) Lim+&+=P+ 1z#O so &im DNE to +42 xrab Oimcf(x=cL @LimF+(xy)=L+Lz Oum(fg)(xy)=LL mfx=/
Methods for computing Imaits
s f(q(x,y>) L5 continuous if g(y) and
xyab
Squeeze
thunuim f(x)=L
If Lim f(x,y)-1,calculate Lim fxx+x=1+1 x10c (01<1) fx+1 12+1
Ex5 Calculate Lim a(01
yz, x+y- are.confinuous f is continuous @ (o,D @o0+1=1 lim =fo= (ay=(01 0+1 9 CalculateUm sin(xy)sin(+y) xy60 goes to0 9005t00 since-1=sinx+y-(sinxy1=sin(rysinxy/sin(xy Since. Lim sin(xy)=sin(a)-O
001'x)
Ex7
<1x1
-1x| Xy JFy
1x1
=1x+ x2+y2 1v2+v
1x2+y
1y1=vyx+y
Lim 1x1=O (xy0,0
lim
FO