Mathematics I · Week 7 — Limits and continuity

Week map

Limit notation

• $\lim_{x \to a} f(x) = L$ → “limit of $f(x)$ as $x$ approaches $a$ is $L$” → value $f$ tends toward near $a$, not necessarily at $a$.
• $\lim_{x \to a^-} f(x)$ → left-hand limit → approach from values less than $a$.
• $\lim_{x \to a^+} f(x)$ → right-hand limit → approach from values greater than $a$.
• $\lim_{x \to a} f(x)$ exists iff left and right limits exist and are equal.

Sequence limits

• $\lim_{n \to \infty} a_n = L$ → terms $a_n$ get arbitrarily close to $L$ as $n$ grows.
• For rational sequences, divide numerator and denominator by highest power of $n$ to guess limit.

Continuity notation

• Continuous at $a$: $\lim_{x \to a} f(x) = f(a)$ — limit exists, function defined, and they agree.
• Removable discontinuity: limit exists but $f(a)$ missing or wrong — “hole” in graph.
• Jump discontinuity: left and right limits exist but differ.
• Infinite discontinuity: limit blows up to $\pm\infty$.

Pattern families

Easy — Sequence pattern

Medium — One-sided and two-sided limits

Hard — Continuity classification

Worked mini-examples

Traps

• Assuming $\lim_{x \to a} f(x) = f(a)$ without checking.
• Ignoring one-sided limits at piecewise boundaries.
• Thinking limit at infinity means function equals that value.
• Canceling $(x - a)$ without noting hole at $x = a$.
• Sequence limit confused with finite term value.

Diagnostic (try yourself)

1. Find $\lim_{x \to 5} (x^2 - 3x)$.
2. Evaluate $\lim_{x \to 1} \frac{x^2 - 1}{x - 1}$.
3. For $f(x) = \frac{|x|}{x}$, what are the left and right limits at $x = 0$?
4. Sequence $a_n = \frac{2n}{n + 5}$: what value does $a_n$ approach as $n \to \infty$?
5. A function has $\lim_{x \to 2} f(x) = 7$ but $f(2) = 3$. Is it continuous at 2? What type of issue is this?