Problem 60 - Thomas Calculus

Question

Show that the approximation of \(\tan x\) by its linearization at the origin must improve as \(x \rightarrow 0\) by showing that . $$ \lim _{x \rightarrow 0} \frac{\tan x}{x}=1 $$

Step-by-Step Solution

Answer

The limit is 1, confirming the approximation improves as \( x \to 0 \).

Step 1: Understand the linearization

The linearization of a function \( f(x) \) at a point \( a \) is given by \( f(a) + f'(a)(x-a) \). For \( \tan x \) at the origin (\( x = 0 \)), the linearization simplifies to \( x \) because \( \tan 0 = 0 \) and \( \sec^2 0 = 1 \).

Step 2: Write the ratio to be evaluated

The problem requires us to find the limit of \( \frac{\tan x}{x} \) as \( x \to 0 \). For small \( x \), \( \tan x \approx x \), so we expect this ratio to approach 1.

Step 3: Use Taylor series expansion

Expand \( \tan x \) using its Taylor series: \( \tan x = x + \frac{x^3}{3} + \frac{2x^5}{15} + \cdots \). For small \( x \), higher-order terms become negligible, giving us \( \tan x \approx x \).

Step 4: Simplify the ratio

Substitute the expansion into the ratio: \( \frac{\tan x}{x} = \frac{x + \frac{x^3}{3} + \cdots}{x} = 1 + \frac{x^2}{3} + \cdots \) which approaches 1 as \( x \rightarrow 0 \).

Step 5: Calculate the limit

Apply the limit directly: \[\lim_{x \to 0} \left( 1 + \frac{x^2}{3} + \cdots \right) = 1\]The limit of higher-order terms is zero as \( x \to 0 \).

Step 6: Conclude the improvement

Since the limit \( \lim_{x \rightarrow 0} \frac{\tan x}{x}=1 \), the approximation of \( \tan x \) by \( x \) gets better as \( x \) approaches zero.

Key Concepts

Linearization

Linearization is a technique used to approximate the value of a function near a certain point. It simplifies complex functions by using straight lines (linear functions).
For a function \( f(x) \), the linear approximation near a point \( a \) is given by:\[ f(a) + f'(a)(x-a) \]

• \( f(a) \) is the function value at the point \( a \).
• \( f'(a) \) is the derivative of the function at the point \( a \), indicating the slope of the tangent line.

In the context of \( \tan x \) at the origin (\( x = 0 \)), the linearization becomes just \( x \), since \( \tan 0 = 0 \) and the derivative \( \sec^2 0 = 1 \).
This linear function, \( x \), serves as a close approximation to \( \tan x \) when \( x \) is near zero, which helps us understand how \( \tan x \) behaves in that vicinity.

Taylor Series

A Taylor series is a powerful tool for approximating functions.
It expresses a function as an infinite sum of terms, calculated from the derivatives of the function at a single point.
The Taylor series for a function \( f(x) \) centered at \( a \) is:\[ f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots\]This series provides insight into how functions like \( \tan x \) can be approximated.
For \( \tan x \), the series expansion looks like:\[ \tan x = x + \frac{x^3}{3} + \frac{2x^5}{15} + \cdots\]

• The first term \( x \) represents the linear approximation at zero.
• Higher-order terms (like \( \frac{x^3}{3} \)) become very small when \( x \) is near zero, refining the approximation.

Taylor Series hence allows for highly accurate approximations when close to the point of expansion, especially beneficial for trigonometric functions in limits and other calculus applications.

Trigonometric Functions

Trigonometric functions play a crucial role in mathematics, especially for describing periodic phenomena.

• Functions such as \( \sin x \), \( \cos x \), and \( \tan x \) are fundamental.
• These functions are periodic, meaning they repeat values in regular intervals.

The toolset provided by calculus, including derivatives, limits, and series expansions, allows us to break down the complexities of trigonometric functions.
When analyzing \( \tan x \) near \( x = 0 \), understanding of limits and infinitesimally small behavior becomes essential.
This is because standard trigonometric functions can be approximated by simpler components (like \( x \) and higher powers) to investigate their behavior in fine detail.
When approaching limit questions such as \( \lim_{x \to 0} \frac{\tan x}{x} = 1 \), recognizing that \( \tan x \approx x \) for small \( x \) is impactful, showcasing the blend of geometry and algebra that trigonometric functions encapsulate.