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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$\begingroup$Direct substitution works for limits only if the value of the function at the limiting point is defined, in other words that there is no discontinuity at that point. In fact the very definition of continuity for a two variable function is that the limit is just equal to the value when substituted:
A function $f(x,y)$ is continuous at $(a,b)$ if $$\lim_{(x,y)\to(a,b)}f(x,y)=f(a,b).$$
Graphically this has the same meaning as in single variable calculus, i.e. that the function doesn't have holes or jumps.
Otherwise, direct substitution fails and you should use other techniques such as employing polar coordinates.
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