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 f (x, y), where C1 is the curve y = x

(x,y)→C1 (0,0)

(b) lim f (x, y), where C2 is the curve y = 2x.

(x,y)→C2 (0,0)

(c) lim f (x, y), where C3 is the curve y = x 2

(x,y)→C3 (0,0)

(d) lim f (x, y).

(x,y)→(0,0)

Answer 1

Given, the function f:R²$\to$ R defined by $f(x, y) = \dfrac{x2 + y}{y}.$

(a)

$\lim_{(x,y) \to (0,0)} f(x,y)= \lim_{(x,y) \to (0,0)}\dfrac{x^2+y}{y} \hspace{1cm}c1:y=x\hspace{0.1cm} \text{is the curve.}\newline\\[9pt] \hspace{3cm}=\lim_{x\to 0}\dfrac{x^2+x}{x} \hspace{1cm}\text{Substitude y=x.}\newline\\[9pt] \hspace{3cm}=\lim_{x\to 0}x+1\newline \hspace{3cm}=1$

(b)

$\lim_{(x,y) \to (0,0)} f(x,y)= \lim_{(x,y) \to (0,0)}\dfrac{x^2+y}{y}\hspace{1cm}c2:y=2x\hspace{0.1cm} \text{is the curve.}\newline\\[9pt] \hspace{3cm}=\lim_{x\to 0}\dfrac{x^2+(2x)}{2x} \hspace{1cm}\text{Substitude y=2x.}\newline\\[9pt] \hspace{3cm}=\lim_{x\to 0}\dfrac{x+2}{2}\newline \hspace{3cm}=1$

(c)

$\lim_{(x,y) \to (0,0)} f(x,y)= \lim_{(x,y) \to (0,0)}\dfrac{x^2+y}{y} \hspace{1cm}c3:y=x^2\hspace{0.1cm} \text{is the curve.}\newline\\[9pt] \hspace{3cm}=\lim_{x\to 0}\dfrac{x^2+(x^2)}{x^2} \hspace{1cm}\text{Substitute}\hspace{0.1cm}y=x^2 .\newline\\[9pt] \hspace{3cm}=\lim_{x\to 0}x^2\newline \hspace{3cm}=0$

(d)

Let $y=mx$ be the curve, where m is a constant.

$\lim_{(x,y) \to (0,0)} f(x,y)= \lim_{(x,y) \to (0,0)}\dfrac{x^2+y}{y} \hspace{1cm}c4:y=mx\hspace{0.1cm} \text{is the curve.}\newline\\[9pt] \hspace{3cm}=\lim_{x\to 0}\dfrac{x^2+(mx)}{mx} \hspace{1cm}\text{Substitute}\hspace{0.1cm}y=mx .\newline\\[9pt] \hspace{3cm}=\lim_{x\to 0}\dfrac{x}{m}+1\newline\\[9pt] \hspace{3cm}=0$