Limit Definitions

A limit describes the value a function approaches near an input. It depends on nearby behavior, so the value of the function at the input may be different or may not exist.

Finite Limit at a Finite Point

The statement

$$\lim_{x \to a} f(x)=L$$

means that $f(x)$ can be made arbitrarily close to $L$ by taking $x$ sufficiently close to $a$, while requiring $x\ne a$.

Formally, for every $\varepsilon>0$, there is a $\delta>0$ such that

$$0<|x-a|<\delta \quad\Longrightarrow\quad |f(x)-L|<\varepsilon.$$

The quantifier order matters: after a desired output tolerance $\varepsilon$ is chosen, one must find an input tolerance $\delta$ that works for every eligible $x$.

One-Sided Limits

The notation

$$\lim_{x\to a^-}f(x)=L \qquad\text{and}\qquad \lim_{x\to a^+}f(x)=L$$

restricts $x$ to values below or above $a$, respectively. This distinction is essential at jumps, endpoints, and piecewise boundaries.

Limits at Infinity

The statement

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

means that $f(x)$ approaches $L$ as $x$ becomes arbitrarily large. Formally, for every $\varepsilon>0$, there is a threshold $M$ such that

$$x>M \quad\Longrightarrow\quad |f(x)-L|<\varepsilon.$$

For $x\to-\infty$, the corresponding condition applies when $x<-M$.

Infinite Limits

The notation

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

means that $f(x)$ eventually exceeds every positive bound as $x$ approaches $a$. It describes unbounded growth rather than a finite real-number limit. The notation $-\infty$ similarly describes values decreasing without bound.