# Why does direct substitution work for limits?

I see a lot of calculus texts stating direct substitution is a form of evaluation for a limit. Maybe I'm missing something because, to me, direct substitution only shows the value of a function $f(x,y)$ for a given value of $(x,y)$. Can we necessarily assume that the limit of the function around $(x,y)$ also converges to that value?

Maybe I need to see a proof to understand if someone can point me in the right direction.

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the crux of it is that it depends on continuity. by definition, a continuous function's limit at $(a,b)$ is just the value of the function at that point. once you've deduced continuity (or manipulated the formula to the point where it's evident), you can just substitute the numbers in – citedcorpse Jun 11 '13 at 10:47
This is valid precisely if $f$ is continuous. – Hagen von Eitzen Jun 11 '13 at 10:48

A function $f(x,y)$ is continuous at $(a,b)$ if $$\lim_{(x,y)\to(a,b)}f(x,y)=f(a,b).$$