Question

Consider the R^2 - R function f defined by f(x,y) = (x^2 +y)/y.

Determine each of the following limits,if it exists.

a) lim (x,y) -> (0,0) f(x,y),where C1 is the curve y=x

b) lim (x,y) -> (0,0) f(x,y),Where C2 is the curve y=2x

c) lim (x,y) -> (0,0) f(x,y),Where C3 is the curve y=x^2

d) lim (x,y) -> (0,0) f(x,y)

Answer 1

ANSWER

a)Let $(x,y)\in C_{1}$ . Substituting $y=x$ into $f(x,y)$ , we have $f(x,x)=\frac{x^{2}+x}{x}$

$\lim_{(x,y)\underset{C_{1}} {\rightarrow}(0,0) }\frac{x^{2}+y}{y}=\lim_{x\rightarrow 0}\frac{x^{2}+x}{x}=\lim_{x\rightarrow 0}(x+1)=1$

b)Let $(x,y)\in C_{2}$ . Substituting $y=2x$ into $f(x,y)$ , we have $f(x,2x)=\frac{x^{2}+2x}{2x}$

$\lim_{(x,y)\underset{C_{2}} {\rightarrow}(0,0) }\frac{x^{2}+y}{y}=\lim_{x\rightarrow 0}\frac{x^{2}+2x}{2x}=\frac {1}{2}\lim_{x\rightarrow 0}(x+2)=1$

c)Let $(x,y)\in C_{3}$ . Substituting $y=x^{2}$ into $f(x,y)$ , we have $f(x,x^{2})=\frac{x^{2}+x^{2}}{x^{2}}=2 (x\neq 0)$

$\lim_{(x,y)\underset{C_{3}} {\rightarrow}(0,0) }\frac{x^{2}+y}{y}=\lim_{x\rightarrow 0}\frac{x^{2}+ x^{2}}{x^{2}}= \lim_{x\rightarrow 0 }2=2$

Note : $\lim_{(x,y)\underset{C_{2}} {\rightarrow}(0,0) }\frac{x^{2}+y}{y}\neq \lim_{(x,y)\underset{C_{3}} {\rightarrow}(0,0) }\frac{x^{2}+y}{y}$ .

Conclusion:

d) $\lim_{(x,y) {\rightarrow}(0,0) }\frac{x^{2}+y}{y}$ does not exist