Question

1. Determine if the following limits exist. In each case prove and explain your argument. (c) lim x +y + y sin x siny *(0,0) XY lim x-(0,0) x4 + y2 lim x+(0,0) x2y2 + (x + y2)2

Answer 1

can lim xy X+ (0,0) 00'te tale are going to n approach co,o), along the x-mois im Zylimo ?10) 4+360) to -01) 027(0,0) 3244344 (200)-1(0,0) s(+300 Along the synowis, lim oc?y? lim ? *60 =0 X 100) alt y4 J oy) och tya - ot y al Then we will use the path y=x, Along this path we have, lim Py2 lim | De lim syn X-7 (0,0) x4 + y4 (a,0) x'tach . x-) (0,0) Jch (2) ? to from 1), (2) and (3) two path gives different values for the limit so the limit doesn't 1-50 C ita prist. 1 aly? (b) lim to so x4 (og x=(0,0) 3 + coc-yje along the w-orais, o lim ozýv alim pa bły? + (x+y)2 (4099100) (0) + (O2)2 = along the Yoonis, lim azy? X5100) 33 + (x-3)2 (0,4)=2190) 0+(-3)2 -(2) along the path y=0 lima ?y2 (0, 1) = (0,0) ? ?+ (8-972 18,2)-1(90) x 3²+ (0-8872 (o ya = lim occ? 6 = lim slim g e 4 6760.7990924432 233 +34

22 17(x-42/2 =lim 13 Lilim sc?y? X-1190) x²y²+(8-4) 2 (2,2)-110,0) 2 (20²+1-20) - from (1), (2) and (3) 1 for two different, have the some value of the limit, so the limit exist

(c) Tim x+100) lim atytay sinasingan 3(90) y . . (1062) along the xronis xty tacy sinsiny - lim xtoto sinasino 60,0)=10,07 ! l Along the x oxis, yalue of the limit doesht exist & The limit doesn't exist for the giuen fonction xy