Intuitive Approach

Understanding Limits: Behavior Near a Point

A limit describes the trend of a function's values as the input approaches a point, not the value at that point itself. When we write $\lim_{x \to a} f(x) = L$, we mean that as $x$ gets arbitrarily close to $a$ (but $x \ne a$), the values $f(x)$ cluster around $L$. Critically, $f(a)$ may not exist, may be undefined, or may differ completely from $L$ — none of this affects the limit.

Estimating Limits from Tables and Graphs

One practical approach is to compute $f(x)$ for values on both sides of $a$, approaching it ever closer. For example, with $f(x) = \frac{x^2 - 1}{x - 1}$ near $x = 1$:

Both sides approach 2, so $\lim_{x\to 1} f(x) = 2$, even though $f(1)$ is undefined.

When Limits Fail to Exist

A two-sided limit exists if and only if the left-hand and right-hand limits are equal: $\lim_{x\to a^-} f(x) = \lim_{x\to a^+} f(x) = L \quad \Leftrightarrow \quad \lim_{x\to a} f(x) = L$

If one-sided limits disagree, the limit does not exist. For instance, the piecewise function with $f(x) = x + 1$ for $x < 2$ and $f(x) = x - 1$ for $x \ge 2$ has $\lim_{x\to 2^-} f(x) = 3$ but $\lim_{x\to 2^+} f(x) = 1$, so the two-sided limit fails to exist.

Limits also fail when $f(x)$ grows without bound or oscillates wildly. For $f(x) = \sin(1/x)$ as $x \to 0$, the function bounces between $-1$ and $1$ infinitely fast; it never settles on any single value, so the limit does not exist.