## University Calculus: Early Transcendentals (3rd Edition)

The informal definition of limit: Suppose that $f(x)$ is defined on an open interval around $c$, but possibly not in $c$ itself. We say that the limit of $f(x)$ as $x$ approaches $c$ is $L$: $$\lim_{x\to c}f(x)=L$$ if for all $x$ sufficiently close to $c$, $f(x)$ is arbitrarily close to $L$.
The informal definition of limit: Suppose that $f(x)$ is defined on an open interval around $c$, but possibly not in $c$ itself. We say that the limit of $f(x)$ as $x$ approaches $c$ is $L$: $$\lim_{x\to c}f(x)=L$$ if for all $x$ sufficiently close to $c$, $f(x)$ is arbitrarily close to $L$. This definition, as said, is informal and might lead to misunderstanding because of imprecise phrases like "sufficiently close" or "arbitrarily close", both of which depend substantially on context. Normal calculations might understand "close" to be within a centimeter or a millimeter even, but in terms of astronomy for example, "close" might mean millions and millions of kilometers. The word "arbitrarily" is also not clear enough. Consider this function: $$f(x)=x$$ As $x\to1$, $f(x)$ not only approaches $1$, but also approaches $2$, $3$, $0$ and a lot of other numbers. Yet we know $\lim_{x\to1}f(x)=1$. So we need a more precise definition to tackle this matter.