# Nakafa Framework: LLM

URL: /en/subject/high-school/12/mathematics/limit/limit-of-trigonometric-function
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Output docs content for large language models.

---

export const metadata = {
   title: "Limit of Trigonometric Function",
   description: "Master trigonometric limits using fundamental theorems. Learn sin x/x equals 1, ratio properties, identities, and advanced substitution techniques.",
   authors: [{ name: "Nabil Akbarazzima Fatih" }],
   date: "05/26/2025",
   subject: "Limits",
};

## Understanding Limits of Trigonometric Functions

Imagine you observe a clock pendulum swinging very slowly approaching its equilibrium point. This motion is similar to the behavior of trigonometric functions when their variable approaches a certain value. **Limits of trigonometric functions** examine how the values of sine, cosine, and tangent functions behave when the input approaches critical points.

Unlike limits of algebraic functions that can often be solved by direct substitution, trigonometric functions have special characteristics due to their periodic and oscillating nature. This makes us need to use **theorems and special properties** to solve trigonometric limits.

## Fundamental Theorem of Trigonometric Limits

The most important foundation in trigonometric limits is the theorem stating that the sine function approaches its gradient when the angle approaches zero.

### Basic Sine Limit

The most fundamental theorem is:

<BlockMath math="\lim_{x \to 0} \frac{\sin x}{x} = 1" />

This theorem **cannot be proven** using direct substitution because it results in the form <InlineMath math="\frac{0}{0}" />. Its proof requires a geometric approach using the unit circle and the squeeze theorem.

> In this theorem, <InlineMath math="x" /> must be in **radians**, not degrees. If using degrees, the result will be different.

### Consequences of the Basic Limit

From the fundamental limit above, we can derive several other important limits:

<div className="flex flex-col gap-4">
<BlockMath math="\lim_{x \to 0} \frac{\tan x}{x} = 1 \quad \text{(because } \tan x = \frac{\sin x}{\cos x} \text{ and } \cos x \to 1\text{)}" />
<BlockMath math="\lim_{x \to 0} \frac{1 - \cos x}{x} = 0 \quad \text{(use L'Hôpital or Taylor expansion)}" />
<BlockMath math="\lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2} \quad \text{(identity } 1 - \cos x = 2\sin^2\frac{x}{2}\text{)}" />
</div>

## Properties of Trigonometric Limits

Based on the fundamental theorem, we can build a series of very useful properties:

### Trigonometric Ratios

For constants <InlineMath math="a \neq 0" /> and <InlineMath math="b \neq 0" />, and **all following limits when <InlineMath math="x \to 0" />**:

| Limit | Result | Notes |
|-------|--------|-------|
| <InlineMath math="\lim_{x \to 0} \frac{\sin ax}{bx}" /> | <InlineMath math="\frac{a}{b}" /> | Manipulation from basic theorem |
| <InlineMath math="\lim_{x \to 0} \frac{\tan ax}{bx}" /> | <InlineMath math="\frac{a}{b}" /> | Because <InlineMath math="\tan ax = \frac{\sin ax}{\cos ax}" /> |
| <InlineMath math="\lim_{x \to 0} \frac{\sin ax}{\sin bx}" /> | <InlineMath math="\frac{a}{b}" /> | Combination of two sine limits |
| <InlineMath math="\lim_{x \to 0} \frac{\tan ax}{\tan bx}" /> | <InlineMath math="\frac{a}{b}" /> | Combination of two tangent limits |

### Trigonometric Combinations

From the trigonometric ratio properties, we can derive several trigonometric combination properties:

<div className="flex flex-col gap-4">
<BlockMath math="\lim_{x \to 0} \frac{\sin ax}{\sin bx} = \frac{a}{b}" />
<BlockMath math="\lim_{x \to 0} \frac{\tan ax}{\sin bx} = \frac{a}{b}" />
</div>

## Techniques for Solving Trigonometric Limits

### Substitution and Manipulation Techniques

When facing complex trigonometric limits, we often need to manipulate expressions to use the fundamental theorem.

**Calculate:** <InlineMath math="\lim_{x \to 0} \frac{\sin 2x}{3x}" />

We manipulate to obtain the standard form:

<div className="flex flex-col gap-4">
<BlockMath math="\lim_{x \to 0} \frac{\sin 2x}{3x} = \lim_{x \to 0} \frac{2 \sin 2x}{2 \cdot 3x} = \lim_{x \to 0} \frac{2}{3} \cdot \frac{\sin 2x}{2x}" />
<BlockMath math="= \frac{2}{3} \lim_{x \to 0} \frac{\sin 2x}{2x} = \frac{2}{3} \cdot 1 = \frac{2}{3}" />
</div>

### Trigonometric Identity Techniques

Often we need to use trigonometric identities to simplify expressions. Always ensure the function is defined at the point being approached.

**Calculate:** <InlineMath math="\lim_{x \to \frac{\pi}{4}} \frac{\sin x + \cos x}{\sin x}" />

Since <InlineMath math="\sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \neq 0" />, the denominator is not zero at <InlineMath math="x = \frac{\pi}{4}" />.

Direct substitution:

<div className="flex flex-col gap-4">
<BlockMath math="\lim_{x \to \frac{\pi}{4}} \frac{\sin x + \cos x}{\sin x} = \frac{\sin \frac{\pi}{4} + \cos \frac{\pi}{4}}{\sin \frac{\pi}{4}}" />
<BlockMath math="= \frac{\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = \frac{2 \cdot \frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = \frac{\sqrt{2}}{\frac{\sqrt{2}}{2}} = \sqrt{2} \cdot \frac{2}{\sqrt{2}} = 2" />
</div>

### Techniques for Special Forms

For limits involving the form <InlineMath math="\frac{0}{0}" />, we need special techniques.

**Calculate:** <InlineMath math="\lim_{x \to \frac{\pi}{2}} \frac{\sin^2(x - \frac{\pi}{2})}{x(x - \frac{\pi}{2})}" />

Let <InlineMath math="u = x - \frac{\pi}{2}" />, then when <InlineMath math="x \to \frac{\pi}{2}" />, we have <InlineMath math="u \to 0" /> and <InlineMath math="x = u + \frac{\pi}{2}" />.

<div className="flex flex-col gap-4">
<BlockMath math="\lim_{u \to 0} \frac{\sin^2 u}{(u + \frac{\pi}{2}) \cdot u} = \lim_{u \to 0} \frac{\sin^2 u}{u^2} \cdot \frac{u}{u + \frac{\pi}{2}}" />
<BlockMath math="= \lim_{u \to 0} \left(\frac{\sin u}{u}\right)^2 \cdot \lim_{u \to 0} \frac{u}{u + \frac{\pi}{2}}" />
<BlockMath math="= 1^2 \cdot \lim_{u \to 0} \frac{u}{\frac{\pi}{2} + u} = 1 \cdot \frac{0}{\frac{\pi}{2}} = 0" />
</div>

## Trigonometric Limits with Angle Identities

### Using Sum and Difference Formulas

When dealing with trigonometric functions involving sum or difference of angles, we can use trigonometric identities.

**Calculate:** <InlineMath math="\lim_{t \to 0} \frac{\cot 5t}{\cot 10t}" />

Using the definition of cotangent:

<div className="flex flex-col gap-4">
<BlockMath math="\lim_{t \to 0} \frac{\cot 5t}{\cot 10t} = \lim_{t \to 0} \frac{\frac{\cos 5t}{\sin 5t}}{\frac{\cos 10t}{\sin 10t}} = \lim_{t \to 0} \frac{\cos 5t \sin 10t}{\sin 5t \cos 10t}" />
<BlockMath math="= \lim_{t \to 0} \frac{\cos 5t}{\cos 10t} \cdot \frac{\sin 10t}{\sin 5t} = 1 \cdot \frac{10}{5} = 2" />
</div>

## Exercises

1. Calculate <InlineMath math="\lim_{x \to 0} \frac{\sin 3x}{4x}" />

2. Calculate <InlineMath math="\lim_{x \to 0} \frac{\tan 2x}{\sin 5x}" />

3. Calculate <InlineMath math="\lim_{x \to 0} \frac{1 - \cos 3x}{x^2}" />

4. Calculate <InlineMath math="\lim_{x \to \frac{\pi}{6}} \frac{\sin x - \frac{1}{2}}{x - \frac{\pi}{6}}" />

5. Calculate <InlineMath math="\lim_{x \to 0} \frac{\sin x \cos x}{x}" />

### Answer Key

1. **Solution:**

   Use algebraic manipulation to obtain the standard form:
   
   <div className="flex flex-col gap-4">
   <BlockMath math="\lim_{x \to 0} \frac{\sin 3x}{4x} = \lim_{x \to 0} \frac{3 \sin 3x}{3 \cdot 4x} = \lim_{x \to 0} \frac{3}{4} \cdot \frac{\sin 3x}{3x}" />
   <BlockMath math="= \frac{3}{4} \lim_{x \to 0} \frac{\sin 3x}{3x} = \frac{3}{4} \cdot 1 = \frac{3}{4}" />
   </div>

2. **Solution:**

   Use trigonometric ratio properties:
   
   <div className="flex flex-col gap-4">
   <BlockMath math="\lim_{x \to 0} \frac{\tan 2x}{\sin 5x} = \lim_{x \to 0} \frac{\tan 2x}{2x} \cdot \frac{5x}{\sin 5x} \cdot \frac{2x}{5x}" />
   <BlockMath math="= \lim_{x \to 0} \frac{\tan 2x}{2x} \cdot \lim_{x \to 0} \frac{5x}{\sin 5x} \cdot \frac{2}{5}" />
   <BlockMath math="= 1 \cdot 1 \cdot \frac{2}{5} = \frac{2}{5}" />
   </div>

3. **Solution:**

   Use the identity <InlineMath math="1 - \cos ax = 2\sin^2\frac{ax}{2}" />:
   
   <div className="flex flex-col gap-4">
   <BlockMath math="\lim_{x \to 0} \frac{1 - \cos 3x}{x^2} = \lim_{x \to 0} \frac{2\sin^2\frac{3x}{2}}{x^2}" />
   <BlockMath math="= \lim_{x \to 0} 2 \cdot \left(\frac{\sin\frac{3x}{2}}{\frac{3x}{2}}\right)^2 \cdot \left(\frac{3}{2}\right)^2" />
   <BlockMath math="= 2 \cdot 1^2 \cdot \frac{9}{4} = \frac{9}{2}" />
   </div>

4. **Solution:**

   Let <InlineMath math="h = x - \frac{\pi}{6}" />, then <InlineMath math="x = h + \frac{\pi}{6}" /> and when <InlineMath math="x \to \frac{\pi}{6}" />, we have <InlineMath math="h \to 0" />:
   
   <div className="flex flex-col gap-4">
   <BlockMath math="\lim_{h \to 0} \frac{\sin(h + \frac{\pi}{6}) - \frac{1}{2}}{h}" />
   <BlockMath math="= \lim_{h \to 0} \frac{\sin h \cos\frac{\pi}{6} + \cos h \sin\frac{\pi}{6} - \frac{1}{2}}{h}" />
   <BlockMath math="= \lim_{h \to 0} \frac{\sin h \cdot \frac{\sqrt{3}}{2} + \cos h \cdot \frac{1}{2} - \frac{1}{2}}{h}" />
   <BlockMath math="= \lim_{h \to 0} \frac{\frac{\sqrt{3}}{2}\sin h + \frac{1}{2}(\cos h - 1)}{h}" />
   <BlockMath math="= \frac{\sqrt{3}}{2} \lim_{h \to 0} \frac{\sin h}{h} + \frac{1}{2} \lim_{h \to 0} \frac{\cos h - 1}{h}" />
   <BlockMath math="= \frac{\sqrt{3}}{2} \cdot 1 + \frac{1}{2} \cdot 0 = \frac{\sqrt{3}}{2}" />
   </div>

5. **Solution:**

   Use the identity <InlineMath math="\sin x \cos x = \frac{1}{2}\sin 2x" />:
   
   <div className="flex flex-col gap-4">
   <BlockMath math="\lim_{x \to 0} \frac{\sin x \cos x}{x} = \lim_{x \to 0} \frac{\frac{1}{2}\sin 2x}{x}" />
   <BlockMath math="= \frac{1}{2} \lim_{x \to 0} \frac{\sin 2x}{x} = \frac{1}{2} \lim_{x \to 0} \frac{2 \sin 2x}{2x}" />
   <BlockMath math="= \frac{1}{2} \cdot 2 \lim_{x \to 0} \frac{\sin 2x}{2x} = 1 \cdot 1 = 1" />
   </div>