In the previous section, we talked about a two‑sided limit: the value $f(x)$ approaches as $x$ gets closer to $a$ from both sides. But sometimes a function behaves differently on the left side of $a$ than on the right side. That's where one‑sided limits come in — they look at the behavior from one direction only.
$$ \lim_{x \to a^{\,-}} f(x) = L \quad \text{(left‑hand limit: $x$ approaches $a$ from values less than $a$)} $$
$$ \lim_{x \to a^{\,+}} f(x) = L \quad \text{(right‑hand limit: $x$ approaches $a$ from values greater than $a$)} $$
🎯 The Big Rule
A two‑sided limit $\displaystyle \lim_{x \to a} f(x)$ exists if and only if both one‑sided limits exist and are equal:
If the left‑hand and right‑hand limits are different, the two‑sided limit does not exist (DNE).
📊 Visualizing One‑Sided Limits
Use the interactive graph below. Pay close attention to what happens at $x = 1$ in the third function — the left and right sides don't agree!
Example 1Matching One‑Sided Limits
Consider $f(x) = \dfrac{x^2 - 1}{x - 1}$ for $x \neq 1$ (the graph has a hole at $x = 1$).
From the left (values smaller than 1):
$\displaystyle \lim_{x \to 1^{\,-}} \frac{x^2-1}{x-1} \;=\; \lim_{x \to 1^{\,-}} (x+1) \;=\; 2$
From the right (values larger than 1):
$\displaystyle \lim_{x \to 1^{\,+}} \frac{x^2-1}{x-1} \;=\; \lim_{x \to 1^{\,+}} (x+1) \;=\; 2$
Both sides approach $2$, so $\displaystyle \lim_{x \to 1} f(x) = 2$ — the two‑sided limit exists even though $f(1)$ is undefined!
Example 2Jump Discontinuity (Limit DNE)
Consider the piecewise function:
\[ f(x) = \begin{cases} x + 1, & x < 0 \\ x - 1, & x \ge 0 \end{cases} \]
From the left ($x$ approaches $0$ from below):
$\displaystyle \lim_{x \to 0^{\,-}} f(x) = 0 + 1 = 1$
From the right ($x$ approaches $0$ from above):
$\displaystyle \lim_{x \to 0^{\,+}} f(x) = 0 - 1 = -1$
Since $1 \neq -1$, the two‑sided limit $\displaystyle \lim_{x \to 0} f(x)$ does not exist (DNE).
⚠️ Common Mistake
Students sometimes see a "hole" in the graph and assume the limit doesn't exist. But a hole does not destroy the limit — the limit is about approaching the point, not about the point's value. A jump, however, does destroy the limit because the left and right approaches lead to different destinations.
🔍 Key Takeaways
The left‑hand limit $\displaystyle \lim_{x \to a^{\,-}} f(x)$ approaches $a$ from smaller numbers.
The right‑hand limit $\displaystyle \lim_{x \to a^{\,+}} f(x)$ approaches $a$ from larger numbers.
A two‑sided limit exists only if both one‑sided limits exist and are equal.
Jump discontinuities cause two‑sided limits to not exist (DNE).