Answer to Question #184472 in Calculus for Njabulo

Question

4. (Section 4.5) 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)→C1 (0,0) f (x, y), where C1 is the curve y = x. (2) (b) lim (x,y)→C2 (0,0) f (x, y), where C2 is the curve y = 2x. (2) (c) lim (x,y)→C3 (0,0) f (x, y), where C3 is the curve y = x 2 . (2) (d) lim (x,y)→(0,0) f (x, y).

Answer 1

Given, the function f:R²$\to$ R defined by f(x, y) = x² + y².

(a)

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

(b)

$\lim_{(x,y) \to (0,0)} f(x,y)= \lim_{(x,y) \to (0,0)}x^2+y^2 \hspace{1cm}c2:y=2x\hspace{0.1cm} \text{is the curve.}\newline \hspace{3cm}=\lim_{x\to 0}x^2+(2x)^2 \hspace{1cm}\text{Substitude y=2x.}\newline \hspace{3cm}=\lim_{x\to 0}5x^2\newline \hspace{3cm}=0$

(c)

$\lim_{(x,y) \to (0,0)} f(x,y)= \lim_{(x,y) \to (0,0)}x^2+y^2 \hspace{1cm}c3:y=x^2\hspace{0.1cm} \text{is the curve.}\newline \hspace{3cm}=\lim_{x\to 0}x^2+(x^2)^2 \hspace{1cm}\text{Substitute}\hspace{0.1cm}y=x^2 .\newline \hspace{3cm}=\lim_{x\to 0}x^2+x^4\newline \hspace{3cm}=0$

(d)

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

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

Thus, in all parts limit exists.