A function $ f(x)$ approaches a limit $ A$ as $ x$ approaches $ a$ if, and only if, for each positive number $ \epsilon$ there is another, $ \delta$, such that whenever $ 0 < |x-a| < \delta $ we have $ |f(x) – A| < \epsilon$. That is, when $ x$ is near $ a$ (within a distance $ \delta$ from it), $ f(x)$ is near $ A$ (within a distance $ \epsilon$ from it). In symbols we write $ \lim_{x \rightarrow a} f(x) = A$.

Here are the implications of this definition.

References

[1] David V. Widder. Advanced Calculus. Dover 1989.