# Definition of Limit of a function in 2 variables

In Thomas's Calculus, 11th Edition, the limit of a function in 2 variables in defined as:

We say that a function $$ƒ(x, y)$$ approaches the limit $$L$$ as $$(x, y)$$ approaches $$(x_0, y_0)$$ if for every number $$\epsilon > 0$$, there exists a corresponding number $$\delta$$ such that for all $$(x, y)$$ in the domain of $$f$$, $$|f(x, y) - L| < \epsilon$$ whenever, $$0 < \sqrt{(x - x_0)^2 + (y - y_0)^2} < \delta$$

Can someone help me understand this definition?

I don't understand the significance of "for every number". Is the definition considering large values of $$\delta$$ and $$\epsilon$$ as well?

What I understand is for some very small delta there should be a very small value of $$\epsilon$$.

• It doesn't matter if $\varepsilon$ is very large, but we're primarily interested in what happens when it is small. Your last sentence is completely backward: what we want is that given a small positive number $\varepsilon$ (doesn't matter how small, it can be less than $10^{-1000000000000000000}$) there should be a corresponding $\delta$ such that... Aug 4, 2019 at 10:16

Although we usually think of $$\epsilon$$ and $$\delta$$ as small numbers, the definition of a limit requires such a $$\delta$$ to exist for every $$\epsilon\gt0$$. So, for example we could ask "What $$\delta$$ causes the function to be at most $$1000$$ units from the limit $$L$$?" i.e. $$\epsilon=1000$$. It may be the case that $$\delta$$ remains small for large $$\epsilon$$ but it is also possible that $$\delta$$ becomes large as it depends on $$\epsilon$$.
• So it doesn't matter what the magnitude of $\delta$ and $\epsilon$ is but just that such a $\delta$ should exist for each positive value of $\epsilon$, right? By this definition then, how would we know if the limit doesn't exist? Aug 4, 2019 at 10:42
• If the limit didn't exist then we would not be able to select any $\delta$ for some $\epsilon\gt0$. Aug 4, 2019 at 10:48