Answer:
Step-by-step explanation:
GIven that :
(a) lim (x,y)→(0,0) 5x + y √ 5x + y + 9 − 3
(b) lim (x,y)→(0,0) 2xy x 2 + y 2
We are to evaluate and determine if the limit exist or not.
From (a);
[tex]\lim \limits _{x \to y (0,0) } 5x + y \sqrt{5x} + y + 9 - 3[/tex]
x = 0
y = 0 (Since no presence of indeterminate)
= [tex]} 5(0 )+ y \sqrt{5(0)} + (0) + 9 - 3[/tex]
= 0+0+0 +9 - 3
= 6
Therefore; the limit exist
(b)
[tex]\lim \limits _{x \to y (0,0) } \dfrac{2xy}{x^2+y^2}[/tex]
If y = mx
[tex]\lim \limits _{x \to y (0,0) } \dfrac{2(mx^2)}{x^2+(mx)^2}[/tex]
[tex]\lim \limits _{x \to y (0,0) } \dfrac{2(mx^2)}{x^2+(m^2x^2)}[/tex]
[tex]= \dfrac{2m }{1+m^2}[/tex]
So; the limit depends on the value of m; then the limit does not exist
