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Week 7: Sequences, Limits, Continuity & Tangents

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Week 7: Sequences, Limits, Continuity & Tangents

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2026-09-12

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Mathematics I — Week 7: Sequences, Limits, Continuity & Tangents

IITM’s Week 7 covers sequences and limits, functions of one variable, graphs and tangents, limits of sequences, and limits of functions of one variable.

TL;DR

Week 7 is the conceptual doorway into calculus.

The main shift is:

\text{exact value} \rightarrow \text{behavior near a value}

The highest-value ideas are:

what a sequence is
convergence vs divergence
what a limit means
limits of simple sequences
limits of functions
left-hand and right-hand limits
continuity intuition
tangent as “local slope”
limits as the tool that makes derivatives possible

1. What is a sequence?

A sequence is an ordered list of numbers:

a_1,a_2,a_3,\dots

More formally, a sequence is a function:

a:\mathbb N\to\mathbb R.

So:

a(n)=a_n.

This is a useful connection to Week 1:

a sequence is just a function whose input is a natural number.

2. Examples of sequences

Arithmetic sequence

2,5,8,11,\dots

Each term increases by:

3.

General term:

a_n=2+3(n-1).

So:

\boxed{ a_n=3n-1 }

Geometric sequence

3,6,12,24,\dots

Each term is multiplied by:

2.

General term:

a_n=3\cdot2^{n-1}.

3. Explicit vs recursive sequence

Explicit

Direct formula for the

n

th term:

a_n=2n+1.

Recursive

Each term defined using earlier terms:

a_1=1

a_{n+1}=a_n+2.

Both can describe the same sequence.

4. Why limits appear

Consider:

a_n=\frac1n.

Then:

1,\frac12,\frac13,\frac14,\dots

As:

n\to\infty,

the terms get closer and closer to:

0.

We write:

\boxed{ \lim_{n\to\infty}\frac1n=0 }

5. Limit intuition

When we write:

\lim_{n\to\infty}a_n=L,

we mean:

as $n$ becomes arbitrarily large, the terms $a_n$ become arbitrarily close to $L$ .

The sequence does not need to ever equal

L

Example:

a_n=\frac1n.

No finite term equals:

0.

But the limit is still:

0.

6. Convergent sequence

A sequence is convergent if it approaches a finite limit.

Example:

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

Rewrite:

a_n=2+\frac1n.

As:

n\to\infty,

\frac1n\to0.

Therefore:

\boxed{ \lim_{n\to\infty}a_n=2 }

7. Divergent sequence

A sequence is divergent if it does not approach a finite single value.

Examples:

a_n=n

grows without bound.

And:

a_n=(-1)^n

oscillates:

-1,1,-1,1,\dots

So it does not settle to one number.

8. Limit laws for sequences

If:

a_n\to A

and:

b_n\to B,

then:

a_n+b_n\to A+B

a_n-b_n\to A-B

a_nb_n\to AB

and if:

B\neq0,

then:

\frac{a_n}{b_n}\to\frac AB.

These mirror ordinary algebra.

9. Constant sequence

If:

a_n=c

for every

n

, then:

\boxed{ \lim_{n\to\infty}a_n=c }

10. Important basic limits

These are worth knowing:

\boxed{ \lim_{n\to\infty}\frac1n=0 }

More generally, for:

p>0,

\boxed{ \lim_{n\to\infty}\frac1{n^p}=0 }

11. Polynomial-ratio sequences

Consider:

\frac{3n^2+2n+1}{n^2+5}.

Divide numerator and denominator by:

n^2.

Then:

\frac{3+\frac2n+\frac1{n^2}} {1+\frac5{n^2}}.

As:

n\to\infty,

all reciprocal terms go to zero.

So:

\boxed{ \lim_{n\to\infty} \frac{3n^2+2n+1}{n^2+5} =3 }

12. Degree comparison rule

For:

\frac{P(n)}{Q(n)},

compare polynomial degrees.

If:

\deg P<\deg Q,

then typically:

\boxed{\text{limit}=0}

If:

\deg P=\deg Q,

then:

\boxed{ \text{limit} = \frac{\text{leading coefficient of }P} {\text{leading coefficient of }Q} }

If:

\deg P>\deg Q,

the expression usually grows without bound or requires more sign analysis.

13. Example: smaller numerator degree

\lim_{n\to\infty} \frac{2n+5}{n^2+1}.

Divide by

n^2

\frac{\frac2n+\frac5{n^2}} {1+\frac1{n^2}}.

Everything in the numerator approaches

0

So:

\boxed{0}

14. Example: larger numerator degree

\frac{n^3}{n^2+1}.

For large

n

\frac{n^3}{n^2}\approx n.

Thus it grows without bound:

\boxed{ \lim_{n\to\infty} \frac{n^3}{n^2+1} = +\infty }

15. Geometric sequences and limits

Consider:

a_n=r^n.

If:

|r|<1,

then:

\boxed{ r^n\to0 }

Example:

\left(\frac12\right)^n\to0.

16. Cases for $r^n$

If:

|r|<1

then:

r^n\to0.

If:

r=1

then:

r^n=1.

If:

r>1

then:

r^n\to+\infty.

If:

r=-1

then:

(-1)^n

oscillates.

If:

r<-1

the magnitude grows and signs alternate.

17. Functions of one variable

A function of one variable typically has the form:

y=f(x).

Input:

x

Output:

f(x).

Examples:

f(x)=x^2

f(x)=\sin x

f(x)=\log x

f(x)=e^x.

Week 7 starts asking:

What does the function do as $x$ approaches some value?

18. Function limit

We write:

\boxed{ \lim_{x\to a}f(x)=L }

to mean:

as $x$ gets arbitrarily close to $a$ , $f(x)$ gets arbitrarily close to $L$ .

Importantly, this concerns values near

a

Not necessarily the value exactly at

a

19. Example

Let:

f(x)=x^2.

Then:

\lim_{x\to2}x^2=4.

x

approaches

2

x^2

approaches

4

20. Direct substitution

For many simple continuous functions, you can evaluate a limit by direct substitution.

Example:

\lim_{x\to3}(2x+5).

Substitute:

2(3)+5=11.

Therefore:

\boxed{11}

21. Polynomial limits

Polynomials are continuous everywhere.

Therefore:

\boxed{ \lim_{x\to a}P(x)=P(a) }

Example:

\lim_{x\to2}(x^3-4x+1).

Substitute:

8-8+1=1.

Thus:

\boxed{1}

22. Rational-function limits

For:

\frac{P(x)}{Q(x)},

direct substitution works if:

Q(a)\neq0.

Example:

\lim_{x\to1} \frac{x+2}{x+3}.

Substitute:

\frac3{4}.

So:

\boxed{\frac34}

23. The interesting case: $0/0$

Consider:

\lim_{x\to1} \frac{x^2-1}{x-1}.

Direct substitution gives:

\frac00.

This does not mean the limit is

0

And it does not mean the limit is automatically undefined.

\frac00

is an indeterminate form.

We need more algebra.

24. Factor and simplify

Factor:

x^2-1=(x-1)(x+1).

Thus:

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

For:

x\neq1,

this equals:

x+1.

Therefore:

\lim_{x\to1} \frac{x^2-1}{x-1} = \lim_{x\to1}(x+1) =2.

So:

\boxed{2}

25. Why cancellation is allowed

At exactly:

x=1,

the original expression is undefined.

But a limit asks about:

x\to1,

not necessarily:

x=1.

For every nearby value with:

x\neq1,

the expression equals:

x+1.

So the limit can exist even though the original function is undefined at the target point.

This is a crucial calculus idea.

26. Removable hole intuition

Consider:

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

Its graph is basically:

y=x+1

except for a missing point at:

(1,2).

The graph has a hole there.

Still:

\lim_{x\to1}f(x)=2.

27. Left-hand limit

The left-hand limit is:

\boxed{ \lim_{x\to a^-}f(x) }

The superscript:

-

means approach

a

from values smaller than

a

Example:

x<a.

28. Right-hand limit

The right-hand limit is:

\boxed{ \lim_{x\to a^+}f(x) }

The superscript:

+

means approach

a

from values greater than

a

29. When does the two-sided limit exist?

The limit:

\lim_{x\to a}f(x)

exists if and only if:

\boxed{ \lim_{x\to a^-}f(x) = \lim_{x\to a^+}f(x) }

and both are finite in the ordinary setting.

If the two sides disagree, the two-sided limit does not exist.

30. Piecewise example

Suppose:

f(x)= \begin{cases} 1,&x<0\\ 2,&x\geq0. \end{cases}

Then:

\lim_{x\to0^-}f(x)=1

but:

\lim_{x\to0^+}f(x)=2.

Since:

1\neq2,

we conclude:

\boxed{ \lim_{x\to0}f(x) \text{ does not exist} }

31. Function value vs limit

These are three separate quantities:

f(a)

\lim_{x\to a^-}f(x)

\lim_{x\to a^+}f(x).

And then there is:

\lim_{x\to a}f(x).

Do not automatically assume they are all equal.

32. Example: limit exists but function value differs

Suppose:

f(x)= \begin{cases} x+2,&x\neq1\\ 10,&x=1. \end{cases}

Then:

f(1)=10.

But near

1

f(x)=x+2.

Therefore:

\lim_{x\to1}f(x)=3.

So:

\boxed{ f(1)\neq\lim_{x\to1}f(x) }

A limit does not care about one isolated point.

33. Continuity intuition

A function is continuous at:

x=a

when three conditions hold:

f(a)

exists.

\lim_{x\to a}f(x)

exists.

\boxed{ \lim_{x\to a}f(x)=f(a) }

Intuitively:

there is no break, jump, or hole at that point.

Formal continuity becomes more important around Week 8.

34. Common discontinuities

Typical ways continuity can fail:

Hole

Function is undefined or has wrong value at a point.

Jump

Left and right limits differ.

Vertical asymptote

Function becomes unbounded near a point.

35. Infinite behavior

Consider:

f(x)=\frac1{x^2}.

As:

x\to0,

the function grows arbitrarily large.

We write:

\boxed{ \lim_{x\to0}\frac1{x^2}=+\infty }

This indicates unbounded growth, not a finite real-number limit.

36. Sign matters

Consider:

f(x)=\frac1x.

As:

x\to0^+,

\frac1x\to+\infty.

But:

x\to0^-,

gives:

\frac1x\to-\infty.

Therefore the two sides behave differently.

37. Limits at infinity

We can also ask what a function does as:

x\to\infty

or:

x\to-\infty.

This connects directly to Week 4 end behavior.

Example:

\lim_{x\to\infty}\frac1x=0.

38. Rational functions at infinity

For:

\frac{P(x)}{Q(x)},

degree comparison again becomes useful.

Example:

\lim_{x\to\infty} \frac{3x^2+1}{2x^2-5}.

Divide by:

x^2.

Then:

\frac{3+\frac1{x^2}} {2-\frac5{x^2}} \to \frac32.

Therefore:

\boxed{\frac32}

39. Horizontal asymptote connection

If:

\lim_{x\to\infty}f(x)=L

or:

\lim_{x\to-\infty}f(x)=L,

then:

y=L

may be a horizontal asymptote.

Example:

f(x)=\frac1x

has:

y=0

as a horizontal asymptote.

40. What is a tangent?

A tangent line is, intuitively, the line that best approximates a curve locally near a point.

For a straight line, slope is easy:

m=\frac{y_2-y_1}{x_2-x_1}.

For a curve, slope changes from point to point.

So how do we define slope at just one point?

Limits.

41. Secant line

Take two points on a curve:

(x,f(x))

and:

(x+h,f(x+h)).

The line through them is a secant line.

Its slope is:

\boxed{ \frac{f(x+h)-f(x)}{h} }

where:

h\neq0.

42. Tangent from secants

Now let:

h\to0.

The second point approaches the first.

If the secant slopes approach a finite value, that limiting value is the tangent slope.

So:

\boxed{ m_{\text{tangent}} = \lim_{h\to0} \frac{f(x+h)-f(x)}{h} }

This is essentially the derivative definition that Week 8 will formalize.

43. Example: tangent intuition for $x^2$

Take:

f(x)=x^2.

At:

x=1,

secant slope:

\frac{f(1+h)-f(1)}h.

Compute:

= \frac{(1+h)^2-1}{h}.

Expand:

= \frac{1+2h+h^2-1}{h}

= \frac{2h+h^2}{h}.

For:

h\neq0,

=2+h.

Now:

h\to0.

Therefore:

m_{\text{tangent}}=2.

Week 8 will call this:

f'(1)=2.

44. Local vs global behavior

This distinction is important.

Global behavior

What the function does across its entire domain.

Local behavior

What the function does very close to one point.

The tangent captures local behavior.

Limits make local analysis precise.

45. Limit laws for functions

If:

\lim_{x\to a}f(x)=L

and:

\lim_{x\to a}g(x)=M,

then:

\lim_{x\to a}[f(x)+g(x)] = L+M

\lim_{x\to a}[f(x)-g(x)] = L-M

\lim_{x\to a}f(x)g(x) = LM

and if:

M\neq0,

\lim_{x\to a}\frac{f(x)}{g(x)} = \frac LM.

46. Constant multiple law

For constant

c

\boxed{ \lim_{x\to a}cf(x) = c\lim_{x\to a}f(x) }

47. Powers

If the limit is appropriate:

\lim_{x\to a}[f(x)]^n = \left( \lim_{x\to a}f(x) \right)^n.

This helps explain why direct substitution works for polynomials.

48. Indeterminate forms

You may encounter:

\frac00

or:

\frac{\infty}{\infty}.

These are not answers.

They mean:

more analysis is needed.

Different expressions producing the same indeterminate form can have completely different limits.

49. Example: three $0/0$ cases

Case 1

\lim_{x\to0}\frac{x}{x}=1.

Case 2

\lim_{x\to0}\frac{x^2}{x}=0.

Case 3

\lim_{x\to0}\frac{x}{x^2}

does not have a finite two-sided limit.

All initially resemble:

\frac00.

Therefore:

\boxed{ 0/0\text{ does not determine the limit} }

50. Rationalization

Sometimes square-root limits can be simplified using a conjugate.

Example:

\lim_{x\to0} \frac{\sqrt{1+x}-1}{x}.

Direct substitution gives:

0/0.

Multiply by the conjugate:

\frac{\sqrt{1+x}+1}{\sqrt{1+x}+1}.

Then numerator becomes:

(1+x)-1=x.

So:

\frac{x} {x(\sqrt{1+x}+1)} = \frac1{\sqrt{1+x}+1}.

Now let:

x\to0.

Result:

\boxed{\frac12}

51. Squeeze intuition

Suppose a function is trapped between two others:

g(x)\leq f(x)\leq h(x)

and both outside functions approach the same value:

\lim_{x\to a}g(x) = \lim_{x\to a}h(x) = L.

Then:

\boxed{ \lim_{x\to a}f(x)=L }

This is the squeeze theorem.

Even if it isn't emphasized immediately, the intuition is useful.

52. Formal limit intuition

A rigorous limit statement:

\lim_{x\to a}f(x)=L

means roughly:

however small a tolerance we demand around $L$ , we can force $f(x)$ inside it by taking $x$ sufficiently close to $a$ .

Later mathematics expresses this with:

\varepsilon

and:

\delta.

You likely do not need to obsess over formal epsilon-delta proofs here unless the course explicitly asks for them.

A sequence:

a_n

is a function with integer inputs.

So:

\lim_{n\to\infty}a_n

is conceptually similar to:

\lim_{x\to\infty}f(x).

The input is moving farther and farther out.

This is why IITM places sequences and function limits together.

Common mistake: infinity as a number

Infinity:

\infty

is not an ordinary real number.

Statements like:

\frac1\infty=0

can be useful informal intuition, but the proper statement is:

\lim_{x\to\infty}\frac1x=0.

That distinction becomes important later.

Common mistake: limit means substitute

Direct substitution works often, but it is not the definition of a limit.

When substitution gives:

\frac00,

you need algebraic simplification or another method.

Common mistake: function value equals limit

Not necessarily.

A function can satisfy:

f(a)=10

while:

\lim_{x\to a}f(x)=3.

Limits concern nearby behavior.

Common mistake: one-sided limits

If:

\lim_{x\to a^-}f(x) \neq \lim_{x\to a^+}f(x),

then:

\lim_{x\to a}f(x)

does not exist.

Even if:

f(a)

itself exists.

Common mistake: cancellation at the forbidden point

For:

\frac{(x-1)(x+2)}{x-1},

you can simplify to:

x+2

only for:

x\neq1.

The simplified expression helps calculate the limit, but it does not magically redefine the original function at

x=1

Common mistake: tangent means “touches once”

A tangent is not simply:

a line that touches the curve at one point.

That elementary definition breaks for many curves.

The better calculus definition is:

the limiting position of secant lines / line with derivative slope at the point.

That model scales properly.

60. Useful strategy for finite-point limits

When solving:

\lim_{x\to a}f(x),

try:


text
1. Direct substitution
2. If ordinary number → done
3. If 0/0:
   • factor
   • cancel
   • rationalize
   • simplify
4. If piecewise:
   • compute left limit
   • compute right limit
5. Check whether both sides agree

61. Strategy for limits at infinity

For rational functions:


text
1. Identify highest power in denominator
2. Divide every term by it
3. Use 1/x^k → 0
4. Evaluate remaining expression

Or quickly compare degrees when the pattern is obvious.

62. Formula / concept sheet

Sequence limit

\boxed{ \lim_{n\to\infty}a_n=L }

Basic reciprocal limit

\boxed{ \lim_{n\to\infty}\frac1{n^p}=0, \quad p>0 }

Function limit

\boxed{ \lim_{x\to a}f(x)=L }

Two-sided existence

\boxed{ \lim_{x\to a}f(x) \text{ exists} \iff \lim_{x\to a^-}f(x) = \lim_{x\to a^+}f(x) }

Continuity condition

\boxed{ \lim_{x\to a}f(x)=f(a) }

with both sides existing.

Secant slope

\boxed{ \frac{f(x+h)-f(x)}h }

Tangent slope preview

\boxed{ \lim_{h\to0} \frac{f(x+h)-f(x)}h }

63. Week 7 conceptual map


text
Sequences
│
├── arithmetic/geometric
├── convergence
└── n → ∞
     │
     ▼
Limits
│
├── finite limit
├── divergence
└── infinity behavior
     │
     ▼
Functions
│
├── x → a
├── left-hand limit
├── right-hand limit
└── continuity intuition
     │
     ▼
Secant lines
     │
     h → 0
     ▼
Tangent slope
     │
     ▼
Derivative

That final arrow is exactly where Week 8 goes.

64. What you should actually master

By the end of Week 7, you should be able to:

Interpret a sequence as a function of $n$ .
Find simple explicit sequence formulas.
Understand convergence and divergence.
Compute basic sequence limits.
Analyze rational sequences using degree comparison.
Understand: $\lim_{x\to a}f(x).$
Evaluate simple limits by substitution.
Resolve basic $0/0$ limits by factorization.
Use rationalization for simple radical limits.
Distinguish function value from limit.
Compute left-hand and right-hand limits.
Determine when a two-sided limit exists.
Understand basic continuity.
Analyze simple limits at infinity.
Understand secant slope.
Understand tangent slope as a limit of secant slopes.

If those are comfortable, Week 7 is under control.

Week 8 covers differentiability and derivatives, computing derivatives and L’Hôpital’s rule, tangents and critical points.

Week 7 asked:

\text{“What happens as }h\to0\text{?”}

Week 8 turns that into the central calculus object:

\boxed{ f'(x)= \lim_{h\to0} \frac{f(x+h)-f(x)}h }

You

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Quick Recall & Flashcards

Test your active recall before proceeding to practice problems. Format:

Question :: Answer.

Limit of sequence $\lim_{n\to\infty} \frac{1}{n}$ :: $0$
Limit at infinity of rational function where deg(numerator) < deg(denominator) :: $0$
Limit at infinity of rational function where deg(numerator) = deg(denominator) :: Ratio of leading coefficients $\frac{a_n}{b_n}$
Three conditions for continuity of $f(x)$ at $x = c$ :: 1. $f(c)$ is defined; 2. $\lim_{x\to c} f(x)$ exists; 3. $\lim_{x\to c} f(x) = f(c)$
Two-sided limit $\lim_{x\to c} f(x) = L$ exists iff :: $\lim_{x\to c^-} f(x) = \lim_{x\to c^+} f(x) = L$

Practice & Self-Check

Attempt these diagnostic questions to verify your operational understanding.

Question 1

Evaluate the limit:

\lim_{n \to \infty} \frac{4n^2 - 3n + 1}{2n^2 + 5n - 7}

NAT

3 Unit Assessment

Question 2

What is

\lim_{x \to 3} \frac{x^2 - 9}{x - 3}

NAT

3 Unit Assessment