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Mathematics I · Week 7 — Sequences, limits & continuity

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Mathematics I · Week 7 — Sequences, limits & continuity

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6 min read

2026-08-16

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Now · Week map

Week 7 — sequences, limits & continuity

Quiz 2 scope: Weeks 1–8 per IITM May 2026 foundation courses. Source baseline: IITM BS admissions important-dates calendar · May 2026 cycle. Times on assessments are operational conventions — verify hall ticket.

Part of the Quiz 2 prep system · Weeks 1–8 index · Pattern atlas · Formula chains.

Week map

Sequence terms → limit intuition → one-sided limits → continuity at a point

Classify → Represent → Execute → Trap-check

Recognize: Ask: What three conditions define continuity at $a$ ?
Procedure: Write first several terms from formula. Describe trend: increasing, decreasing, bounded. Guess limit by table or simplifying $a_n$ for large $n$ (e.g. divide numerator and denominator by highest power).
Variations / traps: Watch for: Assuming limit equals function value without checking.

Formula chain (compressed)

sequence aₙ → limit L → one-sided limits → continuity at a point.

Sequence limit — aₙ → L as n→∞ — terms approach L
Left / right limit — lim_{x→a⁻} f, lim_{x→a⁺} f — piecewise or endpoints
Limit exists — left = right = L — two-sided limit at a
Continuity — lim_{x→a} f(x) = f(a) — no jump/hole at a
Standard limits — lim sin x/x = 1 (x→0) — squeeze / known forms

Open interactive formula desk · Week 7 tab.

Deep study

Mathematics I · Week 7 — Limits and continuity

Deep study for Quiz 2 week 7. Limits describe approaching behavior; continuity requires the approach to match the actual value.

Week map

Sequence terms → limit intuition → one-sided limits → two-sided limit → continuity at a point → types of discontinuity.

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.

Mini-example:

f(x) = \frac{x^2 - 1}{x - 1}

for

x \neq 1

. Near

x = 1

: factor to

x + 1

, so

\lim_{x \to 1} f(x) = 2

, even though

f(1)

is undefined.

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.

Mini-example:

a_n = \frac{3n + 1}{n + 2}

. Divide top and bottom by

n

\frac{3 + 1/n}{1 + 2/n} \to 3

n \to \infty

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 $\pm\infty$ .

Mini-example:

f(x) = |x|/x

x = 0

. Left limit

-1

, right limit

+1

— jump discontinuity.

Pattern families

Easy — Sequence pattern

Write first several terms from formula. Describe trend. Guess limit by table or simplifying for large

n

Medium — One-sided and two-sided limits

Evaluate limits from graphs. Identify left vs right behavior at breakpoints, absolute value corners, piecewise joins.

Hard — Continuity classification

Decide if function is continuous at a point. Identify discontinuity type. State what value would make function continuous (removable case).

Worked mini-examples

Example 1 — Direct substitution.

\lim_{x \to 3} (2x + 5) = 11

— polynomial, no issue.

Example 2 — Cancel factor.

\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = \lim_{x \to 2} (x + 2) = 4

Example 3 — One-sided.

f(x) = \begin{cases} x+1 & x < 0 \\ x^2 & x \geq 0 \end{cases}

. Left at 0:

1

; right at 0:

0

. Two-sided limit does not exist.

Example 4 — Sequence.

a_n = \frac{1}{n} \to 0

n \to \infty

Example 5 — Removable.

f(x) = \frac{\sin x}{x}

for

x \neq 0

f(0) = 0

\lim_{x \to 0} \frac{\sin x}{x} = 1 \neq f(0)

— removable if redefine

f(0) = 1

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)

Find $\lim_{x \to 5} (x^2 - 3x)$ .
Evaluate $\lim_{x \to 1} \frac{x^2 - 1}{x - 1}$ .
For $f(x) = \frac{|x|}{x}$ , what are the left and right limits at $x = 0$ ?
Sequence $a_n = \frac{2n}{n + 5}$ : what value does $a_n$ approach as $n \to \infty$ ?
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?

ChatGPT prep archive

Chatty reference archive

Archived import for extra depth — complements the notes above, not official IITM material.

Core concepts

Sequence limit: terms $a_n$ approach $L$ as $n$ grows; not every sequence converges.
Function limit at $a$ : value approached as $x \to a$ (may differ from $f(a)$ ).
One-sided: $\lim_{x \to a^-} f(x)$ and $\lim_{x \to a^+} f(x)$ .
Continuous at $a$ : $\lim_{x \to a} f(x) = f(a)$ ; all three must agree.

Notation & vocabulary

Term	Meaning
$\lim_{n\to\infty} a_n = L$	sequence converges to L
$\lim_{x\to a} f(x)$	two-sided limit
Discontinuity	limit exists but $\neq f(a)$ , or limit fails

Pattern families

Easy — Sequence pattern

Write first several terms from formula. Describe trend: increasing, decreasing, bounded. Guess limit by table or simplifying

a_n

for large

n

(e.g. divide numerator and denominator by highest power).

Medium — Limit by algebra

Factor rational functions to cancel hole-causing terms. For piecewise functions, match one-sided limits at junction. If left

\neq

right, two-sided limit does not exist.

Hard — Continuity classification

Evaluate

f(a)

, left limit, right limit. Removable if limits agree but

f(a)

wrong or missing. Jump if one-sided limits differ. Infinite if unbounded approach.

Drill these on the pattern atlas — filter to week 7.

Traps

Assuming limit equals function value without checking.
Ignoring one-sided limits at piecewise knots.
Sequence index $n$ vs function variable $x$ .
Thinking oscillation implies convergence.

Retrieval prompts

What three conditions define continuity at $a$ ?
When does a two-sided limit fail to exist?
How do you find $\lim_{n\to\infty} \frac{3n+1}{n}$ ?

Practice loop

Read Deep study (if present) or core concepts once.
Recite the formula chain without looking.
Open one easy pattern on the interactive atlas for week 7.
Attempt without solutions; mark studied after an honest try.
Say one trap aloud before closing the tab.