Quiz 3

Week 7: Sequences, Limits, Continuity & Tangents

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2026-09-12T00:00:00.000Z
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Sequences, convergence, limits at infinity, rational degree rules, two-sided limits, continuity, and tangent intuition. # 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 fu...

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:
exact value→behavior near a value\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:
a1,a2,a3,…a_1,a_2,a_3,\dots
More formally, a sequence is a function:
a:N→R.a:\mathbb N\to\mathbb R.
So:
a(n)=an.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,…2,5,8,11,\dots
Each term increases by:
3.3.
General term:
an=2+3(n−1).a_n=2+3(n-1).
So:
an=3n−1\boxed{ a_n=3n-1 }

Geometric sequence

3,6,12,24,…3,6,12,24,\dots
Each term is multiplied by:
2.2.
General term:
an=3⋅2n−1.a_n=3\cdot2^{n-1}.

3. Explicit vs recursive sequence

Explicit

Direct formula for the nnth term:
an=2n+1.a_n=2n+1.

Recursive

Each term defined using earlier terms:
a1=1a_1=1 an+1=an+2.a_{n+1}=a_n+2.
Both can describe the same sequence.

4. Why limits appear

Consider:
an=1n.a_n=\frac1n.
Then:
1,12,13,14,…1,\frac12,\frac13,\frac14,\dots
As:
n→∞,n\to\infty,
the terms get closer and closer to:
0.0.
We write:
lim⁡n→∞1n=0\boxed{ \lim_{n\to\infty}\frac1n=0 }

5. Limit intuition

When we write:
lim⁡n→∞an=L,\lim_{n\to\infty}a_n=L,
we mean:
as nn becomes arbitrarily large, the terms ana_n become arbitrarily close to LL.
The sequence does not need to ever equal LL.
Example:
an=1n.a_n=\frac1n.
No finite term equals:
0.0.
But the limit is still:
0.0.

6. Convergent sequence

A sequence is convergent if it approaches a finite limit.
Example:
an=2n+1n.a_n=\frac{2n+1}{n}.
Rewrite:
an=2+1n.a_n=2+\frac1n.
As:
n→∞,n\to\infty, 1n→0.\frac1n\to0.
Therefore:
lim⁡n→∞an=2\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:
an=na_n=n
grows without bound.
And:
an=(−1)na_n=(-1)^n
oscillates:
−1,1,−1,1,…-1,1,-1,1,\dots
So it does not settle to one number.

8. Limit laws for sequences

If:
an→Aa_n\to A
and:
bn→B,b_n\to B,
then:
an+bn→A+Ba_n+b_n\to A+B an−bn→A−Ba_n-b_n\to A-B anbn→ABa_nb_n\to AB
and if:
B≠0,B\neq0,
then:
anbn→AB.\frac{a_n}{b_n}\to\frac AB.
These mirror ordinary algebra.

9. Constant sequence

If:
an=ca_n=c
for every nn, then:
lim⁡n→∞an=c\boxed{ \lim_{n\to\infty}a_n=c }

10. Important basic limits

These are worth knowing:
lim⁡n→∞1n=0\boxed{ \lim_{n\to\infty}\frac1n=0 }
More generally, for:
p>0,p>0, lim⁡n→∞1np=0\boxed{ \lim_{n\to\infty}\frac1{n^p}=0 }

11. Polynomial-ratio sequences

Consider:
3n2+2n+1n2+5.\frac{3n^2+2n+1}{n^2+5}.
Divide numerator and denominator by:
n2.n^2.
Then:
3+2n+1n21+5n2.\frac{3+\frac2n+\frac1{n^2}} {1+\frac5{n^2}}.
As:
n→∞,n\to\infty,
all reciprocal terms go to zero.
So:
lim⁡n→∞3n2+2n+1n2+5=3\boxed{ \lim_{n\to\infty} \frac{3n^2+2n+1}{n^2+5} =3 }

12. Degree comparison rule

For:
P(n)Q(n),\frac{P(n)}{Q(n)},
compare polynomial degrees.
If:
deg⁡P<deg⁡Q,\deg P<\deg Q,
then typically:
limit=0\boxed{\text{limit}=0}
If:
deg⁡P=deg⁡Q,\deg P=\deg Q,
then:
limit=leading coefficient of Pleading coefficient of Q\boxed{ \text{limit} = \frac{\text{leading coefficient of }P} {\text{leading coefficient of }Q} }
If:
deg⁡P>deg⁡Q,\deg P>\deg Q,
the expression usually grows without bound or requires more sign analysis.

13. Example: smaller numerator degree

lim⁡n→∞2n+5n2+1.\lim_{n\to\infty} \frac{2n+5}{n^2+1}.
Divide by n2n^2:
2n+5n21+1n2.\frac{\frac2n+\frac5{n^2}} {1+\frac1{n^2}}.
Everything in the numerator approaches 00.
So:
0\boxed{0}

14. Example: larger numerator degree

n3n2+1.\frac{n^3}{n^2+1}.
For large nn:
n3n2≈n.\frac{n^3}{n^2}\approx n.
Thus it grows without bound:
lim⁡n→∞n3n2+1=+∞\boxed{ \lim_{n\to\infty} \frac{n^3}{n^2+1} = +\infty }

15. Geometric sequences and limits

Consider:
an=rn.a_n=r^n.
If:
∣r∣<1,|r|<1,
then:
rn→0\boxed{ r^n\to0 }
Example:
(12)n→0.\left(\frac12\right)^n\to0.

16. Cases for rnr^n

If:

∣r∣<1|r|<1
then:
rn→0.r^n\to0.

If:

r=1r=1
then:
rn=1.r^n=1.

If:

r>1r>1
then:
rn→+∞.r^n\to+\infty.

If:

r=−1r=-1
then:
(−1)n(-1)^n
oscillates.

If:

r<−1r<-1
the magnitude grows and signs alternate.

17. Functions of one variable

A function of one variable typically has the form:
y=f(x).y=f(x).
Input:
xx
Output:
f(x).f(x).
Examples:
f(x)=x2f(x)=x^2 f(x)=sin⁡xf(x)=\sin x f(x)=log⁡xf(x)=\log x f(x)=ex.f(x)=e^x.
Week 7 starts asking:
What does the function do as xx approaches some value?

18. Function limit

We write:
lim⁡x→af(x)=L\boxed{ \lim_{x\to a}f(x)=L }
to mean:
as xx gets arbitrarily close to aa, f(x)f(x) gets arbitrarily close to LL.
Importantly, this concerns values near aa.
Not necessarily the value exactly at aa.

19. Example

Let:
f(x)=x2.f(x)=x^2.
Then:
lim⁡x→2x2=4.\lim_{x\to2}x^2=4.
As xx approaches 22, x2x^2 approaches 44.

20. Direct substitution

For many simple continuous functions, you can evaluate a limit by direct substitution.
Example:
lim⁡x→3(2x+5).\lim_{x\to3}(2x+5).
Substitute:
2(3)+5=11.2(3)+5=11.
Therefore:
11\boxed{11}

21. Polynomial limits

Polynomials are continuous everywhere.
Therefore:
lim⁡x→aP(x)=P(a)\boxed{ \lim_{x\to a}P(x)=P(a) }
Example:
lim⁡x→2(x3−4x+1).\lim_{x\to2}(x^3-4x+1).
Substitute:
8−8+1=1.8-8+1=1.
Thus:
1\boxed{1}

22. Rational-function limits

For:
P(x)Q(x),\frac{P(x)}{Q(x)},
direct substitution works if:
Q(a)≠0.Q(a)\neq0.
Example:
lim⁡x→1x+2x+3.\lim_{x\to1} \frac{x+2}{x+3}.
Substitute:
34.\frac3{4}.
So:
34\boxed{\frac34}

23. The interesting case: 0/00/0

Consider:
lim⁡x→1x2−1x−1.\lim_{x\to1} \frac{x^2-1}{x-1}.
Direct substitution gives:
00.\frac00.
This does not mean the limit is 00.
And it does not mean the limit is automatically undefined.
00\frac00
is an indeterminate form.
We need more algebra.

24. Factor and simplify

Factor:
x2−1=(x−1)(x+1).x^2-1=(x-1)(x+1).
Thus:
x2−1x−1=(x−1)(x+1)x−1.\frac{x^2-1}{x-1} = \frac{(x-1)(x+1)}{x-1}.
For:
x≠1,x\neq1,
this equals:
x+1.x+1.
Therefore:
lim⁡x→1x2−1x−1=lim⁡x→1(x+1)=2.\lim_{x\to1} \frac{x^2-1}{x-1} = \lim_{x\to1}(x+1) =2.
So:
2\boxed{2}

25. Why cancellation is allowed

At exactly:
x=1,x=1,
the original expression is undefined.
But a limit asks about:
x→1,x\to1,
not necessarily:
x=1.x=1.
For every nearby value with:
x≠1,x\neq1,
the expression equals:
x+1.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)=x2−1x−1.f(x)=\frac{x^2-1}{x-1}.
Its graph is basically:
y=x+1y=x+1
except for a missing point at:
(1,2).(1,2).
The graph has a hole there.
Still:
lim⁡x→1f(x)=2.\lim_{x\to1}f(x)=2.

27. Left-hand limit

The left-hand limit is:
lim⁡x→a−f(x)\boxed{ \lim_{x\to a^-}f(x) }
The superscript:
−-
means approach aa from values smaller than aa.
Example:
x<a.x<a.

28. Right-hand limit

The right-hand limit is:
lim⁡x→a+f(x)\boxed{ \lim_{x\to a^+}f(x) }
The superscript:
++
means approach aa from values greater than aa.

29. When does the two-sided limit exist?

The limit:
lim⁡x→af(x)\lim_{x\to a}f(x)
exists if and only if:
lim⁡x→a−f(x)=lim⁡x→a+f(x)\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)={1,x<02,x≥0.f(x)= \begin{cases} 1,&x<0\\ 2,&x\geq0. \end{cases}
Then:
lim⁡x→0−f(x)=1\lim_{x\to0^-}f(x)=1
but:
lim⁡x→0+f(x)=2.\lim_{x\to0^+}f(x)=2.
Since:
1≠2,1\neq2,
we conclude:
lim⁡x→0f(x) does not exist\boxed{ \lim_{x\to0}f(x) \text{ does not exist} }

31. Function value vs limit

These are three separate quantities:
f(a)f(a) lim⁡x→a−f(x)\lim_{x\to a^-}f(x) lim⁡x→a+f(x).\lim_{x\to a^+}f(x).
And then there is:
lim⁡x→af(x).\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)={x+2,x≠110,x=1.f(x)= \begin{cases} x+2,&x\neq1\\ 10,&x=1. \end{cases}
Then:
f(1)=10.f(1)=10.
But near 11:
f(x)=x+2.f(x)=x+2.
Therefore:
lim⁡x→1f(x)=3.\lim_{x\to1}f(x)=3.
So:
f(1)≠lim⁡x→1f(x)\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=ax=a
when three conditions hold:
f(a)f(a)
exists.
lim⁡x→af(x)\lim_{x\to a}f(x)
exists.
lim⁡x→af(x)=f(a)\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)=1x2.f(x)=\frac1{x^2}.
As:
x→0,x\to0,
the function grows arbitrarily large.
We write:
lim⁡x→01x2=+∞\boxed{ \lim_{x\to0}\frac1{x^2}=+\infty }
This indicates unbounded growth, not a finite real-number limit.

36. Sign matters

Consider:
f(x)=1x.f(x)=\frac1x.
As:
x→0+,x\to0^+, 1x→+∞.\frac1x\to+\infty.
But:
x→0−,x\to0^-,
gives:
1x→−∞.\frac1x\to-\infty.
Therefore the two sides behave differently.

37. Limits at infinity

We can also ask what a function does as:
x→∞x\to\infty
or:
x→−∞.x\to-\infty.
This connects directly to Week 4 end behavior.
Example:
lim⁡x→∞1x=0.\lim_{x\to\infty}\frac1x=0.

38. Rational functions at infinity

For:
P(x)Q(x),\frac{P(x)}{Q(x)},
degree comparison again becomes useful.
Example:
lim⁡x→∞3x2+12x2−5.\lim_{x\to\infty} \frac{3x^2+1}{2x^2-5}.
Divide by:
x2.x^2.
Then:
3+1x22−5x2→32.\frac{3+\frac1{x^2}} {2-\frac5{x^2}} \to \frac32.
Therefore:
32\boxed{\frac32}

39. Horizontal asymptote connection

If:
lim⁡x→∞f(x)=L\lim_{x\to\infty}f(x)=L
or:
lim⁡x→−∞f(x)=L,\lim_{x\to-\infty}f(x)=L,
then:
y=Ly=L
may be a horizontal asymptote.
Example:
f(x)=1xf(x)=\frac1x
has:
y=0y=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=y2−y1x2−x1.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))(x,f(x))
and:
(x+h,f(x+h)).(x+h,f(x+h)).
The line through them is a secant line.
Its slope is:
f(x+h)−f(x)h\boxed{ \frac{f(x+h)-f(x)}{h} }
where:
h≠0.h\neq0.

42. Tangent from secants

Now let:
h→0.h\to0.
The second point approaches the first.
If the secant slopes approach a finite value, that limiting value is the tangent slope.
So:
mtangent=lim⁡h→0f(x+h)−f(x)h\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 x2x^2

Take:
f(x)=x2.f(x)=x^2.
At:
x=1,x=1,
secant slope:
f(1+h)−f(1)h.\frac{f(1+h)-f(1)}h.
Compute:
=(1+h)2−1h.= \frac{(1+h)^2-1}{h}.
Expand:
=1+2h+h2−1h= \frac{1+2h+h^2-1}{h} =2h+h2h.= \frac{2h+h^2}{h}.
For:
h≠0,h\neq0, =2+h.=2+h.
Now:
h→0.h\to0.
Therefore:
mtangent=2.m_{\text{tangent}}=2.
Week 8 will call this:
f′(1)=2.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→af(x)=L\lim_{x\to a}f(x)=L
and:
lim⁡x→ag(x)=M,\lim_{x\to a}g(x)=M,
then:
lim⁡x→a[f(x)+g(x)]=L+M\lim_{x\to a}[f(x)+g(x)] = L+M lim⁡x→a[f(x)−g(x)]=L−M\lim_{x\to a}[f(x)-g(x)] = L-M lim⁡x→af(x)g(x)=LM\lim_{x\to a}f(x)g(x) = LM
and if:
M≠0,M\neq0, lim⁡x→af(x)g(x)=LM.\lim_{x\to a}\frac{f(x)}{g(x)} = \frac LM.

46. Constant multiple law

For constant cc:
lim⁡x→acf(x)=clim⁡x→af(x)\boxed{ \lim_{x\to a}cf(x) = c\lim_{x\to a}f(x) }

47. Powers

If the limit is appropriate:
lim⁡x→a[f(x)]n=(lim⁡x→af(x))n.\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:
00\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 0/00/0 cases

Case 1

lim⁡x→0xx=1.\lim_{x\to0}\frac{x}{x}=1.

Case 2

lim⁡x→0x2x=0.\lim_{x\to0}\frac{x^2}{x}=0.

Case 3

lim⁡x→0xx2\lim_{x\to0}\frac{x}{x^2}
does not have a finite two-sided limit.
All initially resemble:
00.\frac00.
Therefore:
0/0 does not determine the limit\boxed{ 0/0\text{ does not determine the limit} }

50. Rationalization

Sometimes square-root limits can be simplified using a conjugate.
Example:
lim⁡x→01+x−1x.\lim_{x\to0} \frac{\sqrt{1+x}-1}{x}.
Direct substitution gives:
0/0.0/0.
Multiply by the conjugate:
1+x+11+x+1.\frac{\sqrt{1+x}+1}{\sqrt{1+x}+1}.
Then numerator becomes:
(1+x)−1=x.(1+x)-1=x.
So:
xx(1+x+1)=11+x+1.\frac{x} {x(\sqrt{1+x}+1)} = \frac1{\sqrt{1+x}+1}.
Now let:
x→0.x\to0.
Result:
12\boxed{\frac12}

51. Squeeze intuition

Suppose a function is trapped between two others:
g(x)≤f(x)≤h(x)g(x)\leq f(x)\leq h(x)
and both outside functions approach the same value:
lim⁡x→ag(x)=lim⁡x→ah(x)=L.\lim_{x\to a}g(x) = \lim_{x\to a}h(x) = L.
Then:
lim⁡x→af(x)=L\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→af(x)=L\lim_{x\to a}f(x)=L
means roughly:
however small a tolerance we demand around LL, we can force f(x)f(x) inside it by taking xx sufficiently close to aa.
Later mathematics expresses this with:
ε\varepsilon
and:
δ.\delta.
You likely do not need to obsess over formal epsilon-delta proofs here unless the course explicitly asks for them.

53. Sequence limits and function limits are related

A sequence:
ana_n
is a function with integer inputs.
So:
lim⁡n→∞an\lim_{n\to\infty}a_n
is conceptually similar to:
lim⁡x→∞f(x).\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:
1∞=0\frac1\infty=0
can be useful informal intuition, but the proper statement is:
lim⁡x→∞1x=0.\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:
00,\frac00,
you need algebraic simplification or another method.
Common mistake: function value equals limit
Not necessarily.
A function can satisfy:
f(a)=10f(a)=10
while:
lim⁡x→af(x)=3.\lim_{x\to a}f(x)=3.
Limits concern nearby behavior.
Common mistake: one-sided limits
If:
lim⁡x→a−f(x)≠lim⁡x→a+f(x),\lim_{x\to a^-}f(x) \neq \lim_{x\to a^+}f(x),
then:
lim⁡x→af(x)\lim_{x\to a}f(x)
does not exist.
Even if:
f(a)f(a)
itself exists.
Common mistake: cancellation at the forbidden point
For:
(x−1)(x+2)x−1,\frac{(x-1)(x+2)}{x-1},
you can simplify to:
x+2x+2
only for:
x≠1.x\neq1.
The simplified expression helps calculate the limit, but it does not magically redefine the original function at x=1x=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→af(x),\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

lim⁡n→∞an=L\boxed{ \lim_{n\to\infty}a_n=L }

Basic reciprocal limit

lim⁡n→∞1np=0,p>0\boxed{ \lim_{n\to\infty}\frac1{n^p}=0, \quad p>0 }

Function limit

lim⁡x→af(x)=L\boxed{ \lim_{x\to a}f(x)=L }

Two-sided existence

lim⁡x→af(x) exists  ⟺  lim⁡x→a−f(x)=lim⁡x→a+f(x)\boxed{ \lim_{x\to a}f(x) \text{ exists} \iff \lim_{x\to a^-}f(x) = \lim_{x\to a^+}f(x) }

Continuity condition

lim⁡x→af(x)=f(a)\boxed{ \lim_{x\to a}f(x)=f(a) }
with both sides existing.

Secant slope

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

Tangent slope preview

lim⁡h→0f(x+h)−f(x)h\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:
  1. Interpret a sequence as a function of nn.
  2. Find simple explicit sequence formulas.
  3. Understand convergence and divergence.
  4. Compute basic sequence limits.
  5. Analyze rational sequences using degree comparison.
  6. Understand: lim⁡x→af(x).\lim_{x\to a}f(x).
  7. Evaluate simple limits by substitution.
  8. Resolve basic 0/00/0 limits by factorization.
  9. Use rationalization for simple radical limits.
  10. Distinguish function value from limit.
  11. Compute left-hand and right-hand limits.
  12. Determine when a two-sided limit exists.
  13. Understand basic continuity.
  14. Analyze simple limits at infinity.
  15. Understand secant slope.
  16. Understand tangent slope as a limit of secant slopes.
If those are comfortable, Week 7 is under control.

Next

Week 8 covers differentiability and derivatives, computing derivatives and L’Hôpital’s rule, tangents and critical points.
Week 7 asked:
“What happens as h→0?”\text{“What happens as }h\to0\text{?”}
Week 8 turns that into the central calculus object:
f′(x)=lim⁡h→0f(x+h)−f(x)h\boxed{ f'(x)= \lim_{h\to0} \frac{f(x+h)-f(x)}h }

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Quick Recall & Flashcards

Test your active recall before proceeding to practice problems. Format:
Question :: Answer.
  • Limit of sequence lim⁡n→∞1n\lim_{n\to\infty} \frac{1}{n} :: 00
  • Limit at infinity of rational function where deg(numerator) < deg(denominator) :: 00
  • Limit at infinity of rational function where deg(numerator) = deg(denominator) :: Ratio of leading coefficients anbn\frac{a_n}{b_n}
  • Three conditions for continuity of f(x)f(x) at x=cx = c :: 1. f(c)f(c) is defined; 2. lim⁡x→cf(x)\lim_{x\to c} f(x) exists; 3. lim⁡x→cf(x)=f(c)\lim_{x\to c} f(x) = f(c)
  • Two-sided limit lim⁡x→cf(x)=L\lim_{x\to c} f(x) = L exists iff :: lim⁡x→c−f(x)=lim⁡x→c+f(x)=L\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.
1

Question 1

Evaluate the limit: lim⁡n→∞4n2−3n+12n2+5n−7\lim_{n \to \infty} \frac{4n^2 - 3n + 1}{2n^2 + 5n - 7}.
NAT
3 Unit Assessment
2

Question 2

What is lim⁡x→3x2−9x−3\lim_{x \to 3} \frac{x^2 - 9}{x - 3}?
NAT
3 Unit Assessment

Active Recall Session

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