Summary of "النهايات من الألف إلى الياء للسنة الثانية ثانوي🚀🚀 | شرح مفصل بأسلوب جديد🔥 | مع تطبيقات✅ لكل عنصر"

Overview

Topic: Limits (“النهايات”) for 2nd-year secondary students — complete coverage from A to Z: theory, geometric meaning (asymptotes), common examples, algebraic rules for removing indeterminate forms, exercises and exam tips.
Emphasis: step-by-step techniques required for the baccalaureate exam — sign tables, factorization, Euclidean (polynomial) division, conjugate (rationalization), comparing degrees of polynomials, extracting common factors, and handling absolute value / radical cases.
Recommended resources: Silver Series (Mathematics, 2nd edition), Prof. Nour El‑Din’s YouTube channel, Akasha publications.

Key concepts and definitions

Limit types
- Infinite limits at a finite point: denominator → 0 gives ±∞ (vertical asymptote).
- Finite limits at infinity: function → real number (horizontal asymptote).
- Infinite limits at infinity: function → ±∞ (no horizontal asymptote; leading-term behavior).
- Indeterminate forms: cases that require algebraic work (e.g., 0/0, ∞/∞, 0·∞, ∞ − ∞).
Asymptotes
- Vertical asymptote: x = a when lim f(x) → ±∞ as x → a±.
- Horizontal asymptote: y = L when lim f(x) = L as x → ±∞.
- Oblique (slant) asymptote: y = ax + b when lim [f(x) − (ax + b)] = 0 as x → ±∞.
  - Compute a = lim f(x)/x and b = lim [f(x) − a x] (as x → ±∞).
Geometric interpretation
- Limits describe graph behavior: approach a vertical line, horizontal line, or slanted line.
- One-sided limits and sign tables indicate approach from left/right (±∞).

Rules / operations on limits (important cases)

Sum
- lim(f + g) = lim f + lim g when both limits are finite.
- If one limit is ±∞ and the other finite, result is ±∞ depending on sign.
- +∞ + (−∞) is indeterminate and needs resolution.
Product
- lim(f·g) = (lim f)·(lim g) when both limits are determinate.
- Nonzero finite × ±∞ = ±∞.
- 0·∞ is indeterminate.
Quotient
- lim(f/g) = (lim f)/(lim g) when denominator limit is nonzero finite.
- finite / ±∞ = 0.
- ±∞ / finite (nonzero) = ±∞.
- ±∞/±∞ and 0/0 are indeterminate.
Common indeterminate forms highlighted
- +∞ − ∞, 0·∞, ∞/∞, 0/0 (also 0^0, ∞^0 but not emphasized here).
Always check left and right limits at points of discontinuity (signs may differ).

Detailed methods — step-by-step procedures

Finding vertical asymptotes / limits where denominator = 0
- Determine domain and identify excluded points (denominator = 0).
- Use sign tables for numerator and denominator (factor as needed).
- Substitute values slightly less/greater than the point or use the sign table to decide ±∞ for one-sided limits.
Limits of rational functions as x → ±∞
- Compare degrees: let n = deg(numerator), m = deg(denominator)
  - n < m → limit = 0 (horizontal asymptote y = 0).
  - n = m → limit = ratio of leading coefficients (horizontal asymptote y = a/b).
  - n = m + 1 → oblique asymptote: perform polynomial division or compute a and b.
  - n > m + 1 → no linear asymptote; polynomial-like growth (sometimes higher-degree polynomial asymptote).
- Practical technique: divide numerator and denominator by the highest power of x present.
Finding an oblique asymptote y = ax + b
- Compute a = lim f(x)/x as x → ±∞ (or do polynomial long division).
- Compute b = lim [f(x) − a x] as x → ±∞.
- If both limits are finite, y = ax + b is the oblique asymptote.
Removing indeterminate forms
- Factorization / cancellation: factor numerator and denominator and cancel common (x − a) when 0/0.
- Euclidean (polynomial) division: write f(x) = (ax + b) + remainder/denominator to reveal oblique asymptote and resolve ∞/∞.
- Multiply by the conjugate: for expressions like √(…) − c multiply numerator and denominator by √(…) + c to cancel the root difference.
- Extract common factor: for √(x² + …) − x, factor x² inside the root and use |x| (consider sign for x → +∞ vs x → −∞).
- Divide numerator and denominator by the highest power of x to manage ∞/∞ cases.
Handling radicals and absolute value
- √(x²) = |x|; for x → +∞ use x, for x → −∞ use −x.
- Rationalize numerators containing differences of roots via the conjugate method.
- Split cases when absolute value is present: treat domains x < c and x > c separately for one-sided limits.
Using sign tables
- Build sign tables for factorized numerator and denominator to determine +/− near points and thus whether limits → +∞ or −∞.
- Sign tables are essential at domain endpoints and for vertical asymptotes.
Using derivatives and calculus tools
- Use derivative to find monotonic intervals, critical points, and local extrema.
- Tangent at x0: slope = f′(x0). To find points where tangent has slope m, solve f′(x) = m and compute corresponding points and tangent equations.
Symmetry and centers
- Example tactic: if f(2 − x) + f(x) = constant (e.g., 2), the curve has a center of symmetry at the midpoint (here (1,1)).

Graphical interpretation checklist

For each limit evaluated (at finite points and at infinity):
- Decide if the curve admits a corresponding asymptote (vertical/horizontal/oblique).
- List vertical asymptotes x = a.
- List horizontal/asymptotes y = L.
- List oblique asymptotes y = ax + b.
- Check the sign of f(x) − (asymptote expression) to determine whether the curve lies above or below the asymptote.

Worked example types (illustrative)

f(x) = 1/(x − a) near x → a±: one-sided limits give ∓∞ depending on direction — vertical asymptote.
Tables of values (0.9, 0.99, 0.999, …) to show growth to ±∞ approaching a forbidden point.
Limits where denominator → 0: use sign tables or substitution with slightly smaller/larger values to get ±∞.
Rational functions with equal degrees: limit = ratio of leading coefficients → horizontal asymptote.
Rational function with deg(numerator) = deg(denominator) + 1: polynomial division → oblique asymptote.
Resolving 0/0 by factoring: (x² − 9)/(x − 3) → cancel (x − 3) → limit = 6 as x → 3.
Radical limits using conjugate: e.g., (√(x − 1) − 3)/(x − 10) → multiply by conjugate to resolve 0/0 and find limit 1/6.
√(x² + …) − x: extract |x| and use sign appropriate to ±∞ to compute limit (example yields ±1/2).
Absolute-value in denominator: split domain (x < 2 and x > 2) to remove |x − 2| and compute one-sided limits (vertical asymptote at x = 2).
Using Euclidean division: write f(x) = ax + b + remainder/denominator to find oblique asymptote and study relative position.

Common exam tips and teacher recommendations

Always write the domain (definition set) first and identify excluded points.
Use a calculator for numerical testing when helpful, but rely on algebra for exact limits.
Build sign tables to determine ±∞ and one-sided behavior.
When faced with an indeterminate form, try methods in this order:
1. Factor/cancel
2. Divide by highest power of x
3. Rationalize (conjugate)
4. Extract common factors
5. Polynomial (Euclidean) division
For radicals, always handle |x| correctly (use sign of x depending on the limit direction).
Memorize the common indeterminate forms and their usual resolutions.
Use Euclidean division when numerator degree ≥ denominator degree.
Practice many examples — these topics are common on baccalaureate exams.

Quick checklist of explicit methods (named in the video)

Sign table method for numerator/denominator near excluded points
Factorization / cancellation
Polynomial (Euclidean) division
Multiply by conjugate (rationalization of numerator)
Extract common factor (especially for √(x² + …) forms)
Divide numerator & denominator by highest power of x at infinity
Use derivative for monotony, extrema and tangent slopes
Compute asymptote coefficients: a = lim f(x)/x ; b = lim[f(x) − a x]

References and recommended materials

Silver Series — Mathematics (2nd edition) (textbook with exercises and video links)
Professor Nour El‑Din — YouTube channel (worked examples and lessons)
Akasha publishing / Akasha Library (publisher referenced)
Suggested tools while studying: notebook, pen, calculator