Calculus Lecture Notes

1

Limits: Intuitive Approach

Introduction to Limits Concept

Exploring the question: Is 0.9̄ = 1? Or equivalently, is 1 - 0.9̄ = 0? We define a function f(n) = 0.99...9 with n nines. We can make f(n) as close to 1 as we like by making n large enough.

f(n) = 0.\underbrace{99...9}_{n}

\[f(3) = 0.999\]

\[f(6) = 0.999999\]

0.\bar{9} = \lim_{n \to \infty} f(n) = 1

📷 Number line showing n=1, n=2, n=? approaching 1

📷 Curve showing transition from discrete points to continuous limit

💡 핵심 개념

To be 1/100th from 1, make n = 5
To be 1/1000000th from 1, make n = 131
Can make f(n) arbitrarily close to 1 by choosing n large enough
$0.\bar{9} = \lim_{n \to \infty} f(n) = 1$

Limits from Graphs

Understanding what an open dot means on a graph. We exclude the point (a, L) or f(a). An open dot indicates that a point has no dimension. The limit notation lim(x$\rightarrow$a) f(x) = L means we can make f(x) or y as close to L as we like by making x close enough to a.

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

📷 Graph showing curve approaching point (a, L) with open circle, excluding f(a)

📷 Vertical dotted line at x = a showing approach to limit L

💡 핵심 개념

Open dot excludes (a, L) or f(a)
A point has no dimension
$\lim_{x \to a} f(x) = L$ means we can make f(x) or y as close to L as we like by making x close enough to a

2

One-Sided Limits

Example: Absolute Value Function

Consider the function f(x) = |x|/x where x ≠ 0. This function approaches different values from left and right.

f(x) = \frac{|x|}{x}, \quad x \neq 0

\lim_{x \to 0^-} \frac{|x|}{x} = -1

\lim_{x \to 0^+} \frac{|x|}{x} = 1

\lim_{x \to 0} \frac{|x|}{x}$ DOES NOT EXIST (DNE)

📷 Graph showing f(x) = |x|/x with horizontal line at y = 1 for x > 0 and y = -1 for x < 0

📷 Open circles at (0, 1) and (0, -1)

📷 Table showing x values (1, 2, 3, 1/4, 1/8) all giving |x|/x = 1

📷 Table showing x values (-1, -2, -3, -1/4, -1/8) all giving |x|/x = -1

💡 핵심 개념

Positive x values: |x|/x = 1
Negative x values: |x|/x = -1
Left limit ≠ Right limit, so limit DNE

One-Sided Limits Definition

Need to define one-sided limits to handle cases where function approaches different values from different directions.

\text{LEFT: } \lim_{x \to a^-} f(x) \text{ means } x < a

\text{RIGHT: } \lim_{x \to a^+} f(x) \text{ means } x > a

\lim_{x \to 0^-} \frac{|x|}{x} = -1 \text{ AND } \lim_{x \to 0^+} \frac{|x|}{x} = 1

\lim_{x \to 0} \frac{|x|}{x} \neq \lim_{x \to 0^+} \frac{|x|}{x}

💡 핵심 개념

LEFT limit: x approaches a from values less than a
RIGHT limit: x approaches a from values greater than a
Limit DNE when left limit ≠ right limit

Example: Piecewise Function at x = -1

Example showing a function g(x) with different one-sided limits at x = -1. Notice that g(-1) = 2 does not affect the limits.

\lim_{x \to -1} g(x)$ DNE B/C

\lim_{x \to -1^-} g(x) = 1$ BUT

\lim_{x \to -1^+} g(x) = 2

\[g(-1) = 2\]

📷 Graph showing piecewise function with different branches meeting at x = -1

📷 Left branch approaches y = 1 at x = -1

📷 Right branch approaches y = 2 at x = -1

📷 Filled dot at (-1, 2) showing actual function value

💡 핵심 개념

Function value g(-1) = 2 doesn't affect the limits
Left and right limits differ, so limit DNE
Actual function value is independent of limit behavior

3

Computing Limits

Non-Piecewise Functions: 3 Cases

Three common cases when computing limits for non-piecewise functions: 1) Plug in and done, 2) Plug in and get 0/0 (indeterminate), 3) Plug in and get k/0 where k ≠ 0 (infinite limit)

💡 핵심 개념

Case 1: PLUG IN AND DONE
Case 2: PLUG IN AND GET 0/0
Case 3: PLUG IN AND GET k/0; k ≠ 0

Case 1: Examples - Direct Substitution

Examples where direct substitution works to evaluate the limit.

\lim_{x \to 2} (x^2 + 2x + 1) = 2^2 + 2(2) + 1

\[y = (x+1)^2 = 9\]

\lim_{x \to 0} e^x = e^0 = 1

\lim_{x \to \pi/3} \cot x = \cot^{-1}\sqrt{3} = \frac{\sqrt{3}}{\sqrt{3}/2} = \frac{1}{\sqrt{3}}

📷 Parabola graph showing y = (x+1)² with vertex

📷 Unit circle diagram showing special angles: 0, π/6 (30°), π/4 (45°), π/3 (60°), π/2 (90°)

📷 Sidebar showing possible outputs of sine and cosine: 0, 1/2, √2/2, √3/2, 1

💡 핵심 개념

Direct substitution works when function is continuous at the point
Polynomial limits: just plug in the value
Exponential limits: evaluate directly
Trigonometric limits: use unit circle values
Special angles: 0°, 30° (π/6), 45° (π/4), 60° (π/3), 90° (π/2)

Case 2: Indeterminate Form 0/0

Example showing indeterminate form that requires further analysis.

\lim_{x \to 0} \frac{\sin x}{x^2} = ?

💡 핵심 개념

0/0 is indeterminate form
Requires algebraic manipulation or L'Hôpital's rule
Cannot simply plug in the value

4

Exercises

🎯 연습문제

1. Prove that 0.9̄ = 1 using limits

$f(n) = 0.\underbrace{99...9}_{n}$으로 정의하면

$\lim_{n \to \infty} f(n) = \lim_{n \to \infty} (1 - 10^{-n}) = 1$

따라서 $0.\bar{9} = 1$

🎯 연습문제

2. Determine if lim(x$\rightarrow$0) |x|/x exists

좌극한: $\lim_{x \to 0^-} \frac{|x|}{x} = \frac{-x}{x} = -1$

우극한: $\lim_{x \to 0^+} \frac{|x|}{x} = \frac{x}{x} = 1$

좌극한 ≠ 우극한이므로 극한은 존재하지 않음 (DNE)

🎯 연습문제

3. Evaluate lim(x$\rightarrow$2) (x² + 2x + 1)

직접 대입법 (Case 1):

$\lim_{x \to 2} (x^2 + 2x + 1) = 2^2 + 2(2) + 1$
$= 4 + 4 + 1 = 9$

또는 $(x+1)^2$로 인수분해: $(2+1)^2 = 9$

🎯 연습문제

4. Find lim(x$\rightarrow$π/3) cot x

직접 대입법:

$\lim_{x \to \pi/3} \cot x = \cot(\pi/3)$

$= \frac{\cos(\pi/3)}{\sin(\pi/3)} = \frac{1/2}{\sqrt{3}/2} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$

🎯 연습문제

5. Identify when limits do not exist due to one-sided limits

극한이 존재하지 않는 경우:

1. 좌극한 ≠ 우극한
예: $f(x) = |x|/x$에서 $x \to 0$

2. 조각함수에서 경계점
각 조각이 다른 값에 접근할 때

3. 점프 불연속
함수값이 갑자기 뛰는 경우

🎯 연습문제

6. Classify limit problems into the three cases (direct substitution, 0/0, k/0)

Case 1: 직접 대입
대입해서 바로 값이 나옴
예: $\lim_{x \to 2} x^2 = 4$

Case 2: 0/0 (부정형)
인수분해, 유리화, 로피탈 필요
예: $\lim_{x \to 0} \frac{\sin x}{x}$

Case 3: k/0 (k≠0)
무한대 극한 또는 DNE
예: $\lim_{x \to 0} \frac{1}{x^2} = \infty$