Introducing Limits Graphically and Numerically
Estimate limits from graphs and tables. Understand the informal ε-δ definition conceptually and distinguish between a function's value at a point and its limit at that point.
A lesson of AP Calculus AB, in Limits and Continuity, about 10 minutes.
Introducing Limits
By the end of this lesson, you will be able to describe the concept of a limit intuitively and numerically, evaluate limits from tables and graphs, and identify when a limit does not exist.
The Central Question
What value does $f(x)$ approach as $x$ gets arbitrarily close to some number $a$?
This is the limit: $$\lim_{x \to a} f(x) = L$$
The key insight: the limit is about what happens near $a$, not at $a$. The function doesn't even need to be defined at $a$.
Numerical Exploration
To estimate $\lim_{x\to 2}\frac{x^2-4}{x-2}$ numerically:
| $x$ | $f(x)$ |
|---|---|
| 1.9 | 3.9 |
| 1.99 | 3.99 |
| 1.999 | 3.999 |
| 2.001 | 4.001 |
| 2.01 | 4.01 |
| 2.1 | 4.1 |
As $x \to 2$ from both sides, $f(x) \to 4$. So $\lim_{x\to 2}\frac{x^2-4}{x-2} = 4$.
Note: $f(2)$ is undefined (0/0), but the limit exists and equals 4.
Graphical Exploration
To read a limit from a graph:
- Approach $x = a$ from the left — what y-value are you approaching?
- Approach $x = a$ from the right — what y-value are you approaching?
- If both give the same value $L$: $\lim_{x\to a}f(x) = L$.
Open circles (holes) don't affect the limit — they just show the function isn't defined there.
When Limits Do Not Exist
Case 1: Left and right limits differ (jump discontinuity)
$f(x) = \begin{cases}x+1 & x<2 \ x^2-1 & x\geq 2\end{cases}$
$\lim_{x\to 2^-} = 3$; $\lim_{x\to 2^+} = 3$ ... actually both 3. But if they differ: DNE.
Case 2: Unbounded behavior
$\lim_{x\to 0}\frac{1}{x^2} = +\infty$ — we say the limit is $+\infty$ (which means it does not exist in the finite sense).
Case 3: Oscillation
$\lim_{x\to 0}\sin(1/x)$ — the function oscillates infinitely between −1 and 1; never settles. DNE.