Limit Theorem ( Trigonometry )

Compulsory
Mathematics

Grade 11

Pen Tool:

CLEAR

Lesson 10 Geometric Limits: \(\lim_{x \to 0} \frac{\sin x}{x} = 1\)

By Ghanashyam Adhikari

Geometrical Proof (Step-by-Step with precise label tracking)

Step 1: Construction Matrix Setup

Let us consider a circle with center \(O\) and radius \(OA = OC = 1\) unit.

Let \(\angle COA = x\) radians be a small positive acute angle.

Draw \(CD \perp OA\) meeting \(OA\) at \(D\). Draw a perpendicular tangent line at \(A\) to meet extended line \(OC\) at point \(B\), making \(BA \perp OA\).

Trigonometric Relations:
In right \(\Delta ODC\): \(\sin x = \frac{CD}{OC} = \frac{CD}{1} \implies CD = \sin x\)
In right \(\Delta OAB\): \(\tan x = \frac{BA}{OA} = \frac{BA}{1} \implies BA = \tan x\)

Step 2: Area Boundaries

By analyzing the nesting boundaries of the figure, we observe the area inequality chain:

Area of \(\Delta OCA\) < Area of Sector \(OCA\) < Area of \(\Delta OAB\)

Applying geometric area formulas:

\(\frac{1}{2} \cdot OA \cdot CD < \frac{1}{2} \cdot (1)^2 \cdot x < \frac{1}{2} \cdot OA \cdot BA\)

Step 3: Algebraic Squeeze Transformation

Substitute lengths derived in Step 1 (\(CD=\sin x\), \(BA=\tan x\)):

\(\frac{1}{2}\sin x < \frac{1}{2}x < \frac{1}{2}\tan x \implies \sin x < x < \tan x\)

Divide by \(\sin x\) and take reciprocals fractions configuration layout:

\(\cos x < \frac{\sin x}{x} < 1\)

Step 4: Sandwich Theorem limit

Evaluate boundaries limit operations as \(x \to 0\):

\(\lim_{x \to 0}\cos x < \lim_{x \to 0}\frac{\sin x}{x} < \lim_{x \to 0}1\)

Since \(\cos(0) = 1\), the Sandwich Principle bounds compression yields the result:

\(\lim_{x \to 0} \frac{\sin x}{x} = 1\)

Q.E.D.

Step 1 / 4

Standard Worked Examples (1 to 5)

Practice Exercises Suite (1 to 5)

Interactive Lesson Quizzes Suite (1 to 5)

Interactive Calculus Sandbox Area

Adjust Angle Variable (\(x\)):

Current Value: 0.70 rad

• Height Line segment (\(CD = \sin x\)) = 0.644

• Sector Arc length Layer (\(\text{Arc } CA = x\)) = 0.700

• Tangent segment Line (\(BA = \tan x\)) = 0.842

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Ratio Value \(\frac{\sin x}{x}\) = 0.9200

Rules & Core Formulas Database

Fundamental Identities:

• \(\lim_{x \to 0}\frac{\sin x}{x} = 1\)
• \(\lim_{x \to 0}\frac{\tan x}{x} = 1\)

Derived Forms:

• \(\lim_{x \to 0}\frac{1-\cos x}{x^2} = \frac{1}{2}\)
• \(\lim_{x \to 0}\frac{\sin ax}{bx} = \frac{a}{b}\)

Limit Curve Graph Vector Plots