(a) Can $\sin(x+y)/(x+y)$ be made continuous by suitably defining it at (0,0)?

Question

(b) Can $xy/(x^2 + y^2)$ be made continuous by suitably defining it at (0,0)?

(c) Prove that $f: \mathbb R^2 \to \mathbb R$, $(x, y) \to ye^x + \sin(x) + (xy)^4$ is continuous.

Attempt

(a) let $t = x + y$

$\lim_{(x, y) \to (0,0)} \frac{sin(x+y)}{x+y} = \lim_{t \to 0} \frac{sin(t)}{t} = 1$

is continuous at $0,0$

(b) Using $y = mx$

$\frac{x \cdot mx}{x^2 + (mx)^2} = \frac{m}{1+m^2}$

Not continuous since it's dependent on value of m

(c)

let $y = x, x = 0$

$\lim_{(x, y) \to (0,0)} ye^x + sin(x) + xy^4 = 0$

therefore continuous

Am I right?

score 2 · Accepted Answer · answered Oct 19 '18 at 05:22

(a) You forgot to notice that the function is not defined on the line $x=-y$. But your reasoning outside of this subdomain is correct.

(b) Is correct

(c) You were asked to prove that the function is continuous everywhere, not that it it's limit is defined at $(0,0)$

1 Answers1