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Suppose that $\lim _{(x, y) \rightarrow(3,1)} f(x, y)=6 .$ What can you say about the value of $f(3,1) ?$ What if $f$ is continuous?

We can't say anything about the value of $f(3,1)$ because we don't knowwhether $f$ is continuous at $(3,1)$ or not. However, if $f$ is continuous at$(3,1),$ then $f(3,1)=6$ since the function value must equal to its limit if $f$is continuous.

Calculus 3

Chapter 11

PARTIAL DERIVATIVES

Section 2

Limits and Continuity

Partial Derivatives

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So let's say we have a function in two variables f of X comma. Why and the limit of f of X comma y as X y approaches three comma one, um, is six. So what does that tell us about this function? Well, first we can say that s for long, equal to some devalue eso If we were to, um, plot this on a coordinate plain x y um, I'm not gonna show the Z axis because that's ah free D. It's a little harder to share in a drawing, but basically what this is saying is, if you have the 0.3 common one as, um as arbitrary points X y approach three comma, one from any and all direct, then the value approaches six. So as X comma y approaches three comma one, so does he approach six Bunch? If we were told, um, that f of X comma, why is continuous, um, at the point three comma one. Then you would be able to say not only does the approach sex, but Z equals six ep of X comma line rather f of three com. One equal six because that's the ignition of continuity. If a function is consideration. I'm at a point. The and the and liver function at a poem is equal to, um, the limit of that, um, value at that point, um, so it's continuous equal. Six. If it's not us, we don't know if it's continuous or not. Then we can say f of x comma y approaches six.

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