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Multivariable Calculus

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Limit Of $\sin(x^2+y^2) / (x^2+y^2)$ At The Origin.

MVCALC-G3FTS1

For real $x$ and $y$ such that $(x,y)\ne (0,0)$ let:

$$f(x,y)=\cfrac{\sin(x^2+y^2)}{x^2+y^2}$$

Can $f(0,0)$ be defined in such a way as to make $f$ continuous at the origin $(0,0)$?

A

No. The function does not have a removable discontinuity at the origin because $$\lim_{(x,y)\to(0,0)} f(x,y)$$ does not exist.

B

Yes. Define $f(0,0)=0$.

C

Yes. Define $f(0,0)=1$.