Existence Of Limits

When Does a Limit Fail to Exist?

A limit $\lim_{x \to a} f(x) = L$ exists when $f(x)$ approaches a single, finite value $L$ from both sides as $x$ approaches $a$. When this doesn't happen, we say the limit does not exist (DNE).

The Core Requirement

A two-sided limit exists if and only if both one-sided limits exist and are equal: $\lim_{x\to a^-} f(x) = \lim_{x\to a^+} f(x) = L$

If the left-hand and right-hand limits differ, there is no single value the function approaches — the limit fails.

Three Ways Limits Fail to Exist

1. Disagreeing One-Sided Limits

When left and right limits approach different values, the limit does not exist. For example: $f(x) = \begin{cases} x+1 & x < 2 \\ 4-x & x \ge 2\end{cases}$

As $x \to 2^-$, $f(x) \to 3$; as $x \to 2^+$, $f(x) \to 2$. Since $3 \ne 2$, the limit at $x=2$ does not exist.

2. Unbounded Growth

If $f(x)$ grows without bound (toward $\pm\infty$) as $x \to a$, no finite limit exists. This creates a vertical asymptote.

3. Oscillation

The function may stay bounded but oscillate without settling. Consider $f(x) = \sin(1/x)$ as $x \to 0$: the argument $1/x$ grows huge, causing $\sin(1/x)$ to cycle repeatedly between $-1$ and $1$ infinitely often. The function never approaches a single value, so the limit does not exist.

Key Insight

Remember: a limit exists only when $f(x)$ approaches one specific, finite number consistently from both sides. The function's value at $a$ itself ($f(a)$) is irrelevant — only the behavior near $a$ matters. Disagreement between sides, unbounded growth, or endless oscillation all signal that the limit fails.