1. What is a Limit?
A limit describes the value that a function approaches as the input approaches a particular point. In other words: as \(x\) gets closer to some number \(a\), what value is \(f(x)\) getting close to?
Intuitive Example:
Imagine walking toward a wall but stopping just before touching it. Your distance from the wall gets closer and closer to 0,
but you never actually touch it. That “getting closer” behavior is what a limit captures.
\[
\lim_{x \to a} f(x) = L
\]
This reads: “The limit of \(f(x)\) as \(x\) approaches \(a\) is \(L\).”
Key Points:
- The function doesn’t have to be defined at \(x=a).
- Limits focus on behavior around the point, not necessarily at the point.
2. Examples
Example A · Simple function
Let \( f(x) = 2x + 3 \). Find \(\lim_{x \to 1} f(x)\).
Let \( f(x) = 2x + 3 \). Find \(\lim_{x \to 1} f(x)\).
Direct evaluation works because \(f\) is continuous at \(x=1\):
\[
f(1) = 2(1) + 3 = 5
\]
Therefore, \(\lim_{x \to 1} f(x) = 5\).
Example B · Function undefined at a point
Let \( f(x) = \dfrac{x^{2} - 1}{x - 1} \). Find \(\lim_{x \to 1} f(x)\).
Let \( f(x) = \dfrac{x^{2} - 1}{x - 1} \). Find \(\lim_{x \to 1} f(x)\).
Direct substitution gives \( \tfrac{0}{0} \), which is indeterminate.
Factor numerator:
\[
\frac{(x-1)(x+1)}{x-1} = x+1, \quad x \neq 1
\]
Now evaluate:
\[
\lim_{x \to 1} f(x) = 1 + 1 = 2
\]
Even though \(f\) is not defined at \(x=1\), the limit still exists and equals 2.
Check Yourself
\(\lim_{x \to 2} (3x - 4) = ?\)
Direct substitution: \(3(2) - 4 = 6 - 4 = 2\).
\(\lim_{x \to 3} \dfrac{x^2 - 9}{x - 3} = ?\)
Factor numerator: \(x^2 - 9 = (x-3)(x+3)\). Then:
\[
\frac{(x-3)(x+3)}{x-3} = x+3,\; x \neq 3
\]
So the limit is \(x+3\) evaluated at \(x=3\): \(3 + 3 = 6\).