Definition of limits

Question

We have $\lim_{x\to\infty} x^2 = \infty$.

What would be the precise definitions of $\lim_{x\to c} f(x) = \infty, \lim_{x\to\infty}f(x) = L, \lim_{x\to\infty}f(x) = \infty$ ? How to go about these? I'm not sure where to even start.

Answer 1

We have the following definitions. Note that $\newcommand{\ve}{\varepsilon} \newcommand{\d}{\delta} \newcommand{\abs}[1]{\left| #1 \right|} \forall$ means "for all" and $\exists$ means "there exists".

Here, we assume $c,L$ are finite.

• $\displaystyle \lim_{x\to c} f(x) = \infty \stackrel{\text{def.}}{\iff} (\forall M > 0)(\exists \d > 0) \Big( 0 < \abs{x-c} < \d \implies f(x) > M \Big)$
• $\displaystyle \lim_{x \to \infty} f(x) = L \stackrel{\text{def.}}{\iff} (\forall \ve > 0)(\exists N > 0)(\forall x \ge N) \Big( \abs{f(x) - L} < \ve \Big)$
• $\displaystyle \lim_{x \to \infty} f(x) = \infty \stackrel{\text{def.}}{\iff} (\forall M > 0)(\exists N > 0)(\forall x \ge N) \Big( f(x) > M \Big)$

(Some minor variations on these exist, but are fundamentally equivalent; for instance, I've seen $M$ be limited to just positive integers.)