Maths 1 Week 7: Sequences & Limits

0. Prerequisites

NOTE

What you need to know:

• Functions: Understanding $f(x)$.
• Infinity: Concept of “getting larger and larger”.
• Floor Function: $\lfloor 2.9\rfloor =2$.

Quick Refresher

• Sequence: A list of numbers $a_1,a_2,a_3\dots$.
• Limit: The value a function approaches as $x$ gets closer to a specific point.
• Continuity: Can you draw the graph without lifting your pen?

---

1. Core Concepts

1.1 Limits of Sequences

• Notation: ${\lim }_{n\to \infty }{a}_n=L$.
• Rational Sequences: $a_n=\frac{P(n)}{Q(n)}$.
  • If Top Degree < Bottom Degree $\to$ Limit is 0.
  • If Top Degree > Bottom Degree $\to$ Limit is $\infty$ (Diverges).
  • If Degrees Equal $\to$ Limit is Ratio of Leading Coefficients.

1.2 Limits of Functions

• Left Hand Limit (LHL): Approaching from left ($x\to c^{-}$).
• Right Hand Limit (RHL): Approaching from right ($x\to c^{+}$).
• Existence: Limit exists ONLY if LHL = RHL.

1.3 Continuity

A function $f$ is continuous at $c$ if three conditions meet:

1. $f(c)$ is defined (Point exists).
2. ${\lim }_{x\to c}f(x)$ exists (LHL = RHL).
3. Limit = Function Value ($\lim =f(c)$).

---

2. Pattern Analysis & Goated Solutions

Pattern 1: Rational Limits (The “Degree Hack”)

Context: “Find ${\lim }_{n\to \infty }\frac{3n^2+5n}{4n^2-1}$.”

TIP

Mental Algorithm:

1. Check Degrees: Look at highest power of $n$ on top and bottom.
2. Compare:
   • Top < Bottom $⟹$ Answer 0.
   • Top > Bottom $⟹$ Answer $\infty$.
   • Top = Bottom $⟹$ Answer $\frac{\text{Top Coeff}}{\text{Bottom Coeff}}$.

Example (Detailed Solution)

Problem: Find ${\lim }_{n\to \infty }\frac{12n^3-5n}{3n^3+2n^2}$. Solution:

1. Identify Degrees:
   • Top: $12n^3$ (Degree 3).
   • Bottom: $3n^3$ (Degree 3).
2. Compare: Degrees are equal (3 = 3).
3. Ratio:
   • Take coefficients of $n^3$.
   • Top: 12.
   • Bottom: 3.
   • Ratio: $12/3=4$. Answer: 4.

---

Pattern 2: Continuity with Unknowns

Context: “Find $k$ so that $f(x)$ is continuous at $x=2$.”

TIP

Mental Algorithm:

1. Find LHL: Plug $x=c$ into the “Left” formula ($x<c$).
2. Find RHL: Plug $x=c$ into the “Right” formula ($x>c$).
3. Equate: Set LHL = RHL and solve for $k$.

Example (Detailed Solution)

Problem: $f(x)=\{\begin{array}{ll}kx+1& x\le 2\\ 3x-2& x>2\end{array}$ Find $k$ for continuity at $x=2$. Solution:

1. Find LHL ($x\le 2$):
   • Use $kx+1$.
   • Plug in $x=2$: $2k+1$.
2. Find RHL ($x>2$):
   • Use $3x-2$.
   • Plug in $x=2$: $3(2)-2=6-2=4$.
3. Equate:
   • LHL = RHL.
   • $2k+1=4$.
   • $2k=3$.
   • $k=1.5$. Answer: $k=1.5$.

---

Pattern 3: Floor Function Limits

Context: “Find ${\lim }_{x\to 3^{-}}\lfloor x\rfloor$.”

TIP

Mental Algorithm:

• Right Limit ($k^{+}$): $\lfloor k.001\rfloor =k$.
• Left Limit ($k^{-}$): $\lfloor k-0.001\rfloor =\lfloor k-1.999\rfloor =k-1$.
• Visual: Step down when coming from left.

Example (Detailed Solution)

Problem: Evaluate ${\lim }_{x\to 4^{-}}(\lfloor x\rfloor +x)$. Solution:

1. Analyze Limit: $x\to 4^{-}$ means $x$ is slightly less than 4 (e.g., 3.99).
2. Floor Part:
   • $\lfloor 3.99\rfloor =3$.
3. Algebraic Part:
   • $x$ approaches 4.
4. Combine:
   • $3+4=7$. Answer: 7.

---

3. Practice Exercises

1. Rational Limit: ${\lim }_{n\to \infty }\frac{n^2}{n^3+1}$.
   • Hint: Top (2) < Bottom (3). Limit is 0.
2. Continuity: $f(x)=x^2$ ($x<1$), $kx$ ($x\ge 1$). Find $k$.
   • Hint: $1^2=k(1)⟹k=1$.
3. Floor: ${\lim }_{x\to 2^{+}}\lfloor x\rfloor$.
   • Hint: $\lfloor 2.01\rfloor =2$.

---

🧠 Level Up: Advanced Practice

Question 1: AP/GP Mixed Problem

Problem: Three numbers are in AP. Their sum is 15. If 1, 4, 19 are added respectively, they form a GP. Find numbers. Logic:

1. AP Terms: $a-d,a,a+d$.
2. Sum: $3a=15⟹a=5$. Terms: $5-d,5,5+d$.
3. Add Constants:
   • $(5-d)+1=6-d$.
   • $5+4=9$.
   • $(5+d)+19=24+d$.
4. GP Condition: $b^2=ac$.
   • $9^2=(6-d)(24+d)$.
   • $81=144+6d-24d-d^2$.
   • $81=144-18d-d^2$.
   • $d^2+18d-63=0$.
5. Solve Quadratic: $(d+21)(d-3)=0$. $d=3$ or $d=-21$.
6. Result:
   • If $d=3$: 2, 5, 8.
   • If $d=-21$: 26, 5, -16. Answer: 2, 5, 8 OR 26, 5, -16.

Question 2: Infinite Series Sum

Problem: Find sum of $5+\frac{5}{2}+\frac{5}{4}+\dots$. Logic:

1. Identify: Infinite GP. $a=5,r=1/2$.
2. Condition: $|r|<1$. Yes.
3. Formula: $S=\frac{a}{1-r}$.
4. Calc: $S=\frac{5}{1-0.5}=\frac{5}{0.5}=10$. Answer: 10.

Question 3: Tricky Limit

Problem: Evaluate ${\lim }_{x\to 2}\frac{x^2-4}{\sqrt{x+2}-2}$. Logic:

1. Check Form: $0/0$. Indeterminate.
2. Rationalize Denominator: Multiply by $\frac{\sqrt{x+2}+2}{\sqrt{x+2}+2}$.
3. Simplify:
   • Num: $(x-2)(x+2)(\sqrt{x+2}+2)$.
   • Denom: $(x+2)-4=x-2$.
4. Cancel: Remove $(x-2)$.
   • Limit becomes $(x+2)(\sqrt{x+2}+2)$.
5. Plug in 2: $(2+2)(\sqrt{4}+2)=4(2+2)=16$. Answer: 16.