In Precalculus you read limits off graphs and tables in Estimating Limits from Graphs and Tables. Calculus is built on that one idea, so we start by stating it carefully. A limit is about where a function is heading as $x$ approaches a number, not about what happens at the number itself.
Key idea
Here is a function with a hole. $f(x) = \dfrac{x^2 - 4}{x - 2}$ is undefined at $x = 2$, because the bottom is 0 there. Yet near 2 its values behave very politely:
| $x$ | 1.9 | 1.99 | 1.999 | 2 | 2.001 | 2.01 | 2.1 |
|---|---|---|---|---|---|---|---|
| $f(x)$ | 3.9 | 3.99 | 3.999 | undefined | 4.001 | 4.01 | 4.1 |
From both sides the outputs close in on 4, so $\lim_{x \to 2} f(x) = 4$. The algebra agrees: for $x \ne 2$, $\frac{x^2-4}{x-2} = \frac{(x-2)(x+2)}{x-2} = x + 2$, and $x + 2$ is near 4 when $x$ is near 2.
Why it works
Three situations to tell apart
- The value is missing but the limit exists (a hole), like the example above.
- The value exists but differs from the limit (a point moved off the curve).
- The value exists and equals the limit. This is the common, well-behaved case, which we will call continuity.
Worked example: A table that hides the function
- From the left, the outputs climb: $7.8,\ 7.98,\ 7.998$.
- From the right, the outputs fall: $8.2,\ 8.02,\ 8.002$.
- Both sides squeeze toward the same number, 8. We never needed $g(3)$.
Answer: $\lim_{x \to 3} g(x) = 8$
Worked example: The value is a decoy
- For every $x$ near 1 (but not equal to 1), the rule $f(x) = x^2 + 1$ applies.
- As $x \to 1$, $x^2 + 1 \to 1 + 1 = 2$.
- The separate value $f(1) = 5$ is a single point floating above the curve. It does not change where the curve is heading.
Answer: $\lim_{x \to 1} f(x) = 2$, while $f(1) = 5$.
Common mistake