What is an infinite limit?

Question

$$\lim_{x\to\infty} f(x) = c$$ $$\lim_{x\to c} f(x) = \infty$$ $$\lim_{x\to\infty} f(x) = \infty$$

When someone says, give a delta epsilon proof of an $\text{infinite limit}$, which do they mean? $c$ is constant.

Answer 1

When we write $\lim_{x\to\infty}f(x)=c$ we mean

$$\forall \epsilon>0,\,\exists M>0 \text{ s.t. } x>M \implies |f(x)-c|<\epsilon$$

meaning that $f$ becomes arbitrarily close to $c$ as $x$ becomes large.

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When we write $\lim_{x\to c}f(x)=\infty$ we mean

$$\forall M>0,\,\exists \delta>0 \text{ s.t. } 0<|x-c|<\delta \implies f(x)>M$$

meaning that $f$ becomes arbitrarily large as $x$ approaches $c$.

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When we write $\lim_{x\to \infty}f(x)=\infty$ we mean

$$\forall M>0,\,\exists N>0 \text{ s.t. } x>N \implies f(x)>M$$

meaning that $f$ becomes arbitrarily large as $x$ becomes large.