Limits
1.1 Intro to Limits
Intuition
A limit describes the value that $f(x)$ approaches as $x$ gets close to a point.
$$ \lim_{x \to a} f(x) = L $$
Definition
Let $f$ be defined on an open interval around $a$ (except possibly at $a$). We say $\lim_{x\to a}f(x)=L$ if values of $f(x)$ can be made arbitrarily close to $L$ by taking $x$ sufficiently close to $a$.
Example
For $f(x)=2x+1$, we have $\lim_{x\to 3}f(x)=7$.
One-Sided Limits
- Left-hand limit: $\lim_{x\to a^-}f(x)$
- Right-hand limit: $\lim_{x\to a^+}f(x)$
- Two-sided limit exists when both one-sided limits exist and are equal.
Warning
A function can be undefined at $x=a$ and still have a limit there.
Small Comparison Table
| Expression | Value |
|---|---|
| $\lim_{x\to 2} (x^2)$ | $4$ |
| $\lim_{x\to 2} (3x-1)$ | $5$ |
| $\lim_{x\to 0} \sin(x)/x$ | $1$ |
Pseudocode Approximation
for x in [a-0.1, a-0.01, a+0.01, a+0.1]:
print(f(x))
Exercise
Estimate $\lim_{x\to 4} \sqrt{x}$ from a table of nearby values.