Problem 11 - A Textbook of Differential Calculus

Question

The vlaue of \(\lim _{x \rightarrow 0} \frac{\sqrt{\sec ^{2} \frac{x}{2}-1}}{x}\) is (a) \(\frac{1}{2}\) (b) \(-\frac{1}{2}\) (c) 1 (d) doesn't exist

Step-by-Step Solution

Answer

The value of the limit is \( \frac{1}{2} \). (Option a).

Step 1: Rewrite the Expression Using Trigonometric Identity

Recall the trigonometric identity for secant: \[ \sec \theta = \frac{1}{\cos \theta} \]Apply it to \( \sec^2 \frac{x}{2} \), giving us:\[ \sec^2 \frac{x}{2} = 1 + \tan^2 \frac{x}{2} \].This allows us to rewrite the expression inside the limit as:\[ \sqrt{\sec^2 \frac{x}{2} - 1} = \sqrt{\tan^2 \frac{x}{2}} = \left| \tan \frac{x}{2} \right| \]. However, near \( x = 0 \), \( \tan \frac{x}{2} \approx \frac{x}{2} \), so we approximate as \( \tan \frac{x}{2} \approx \frac{x}{2} \) near zero.

Step 2: Substitute the Trigonometric Approximation

Using the approximation \( \tan \frac{x}{2} \approx \frac{x}{2} \) near \( x = 0 \), substitute into the limit expression:\[ \lim _{x \rightarrow 0} \frac{\sqrt{\tan^2 \frac{x}{2}}}{x} = \lim _{x \rightarrow 0} \frac{\left| \tan \frac{x}{2} \right|}{x} \approx \lim _{x \rightarrow 0} \frac{\left| \frac{x}{2} \right|}{x} \].

Step 3: Simplify the Expression

The expression simplifies to:\[ \lim _{x \rightarrow 0} \frac{\frac{x}{2}}{x} = \lim _{x \rightarrow 0} \frac{1}{2} \].Since the absolute value does not affect the result for \( x \rightarrow 0^+ \) and \( x \rightarrow 0^- \), the limit is unaffected by the direction of the approach to zero.

Step 4: Calculate the Limit

Now calculate the limit:\[ \lim _{x \rightarrow 0} \frac{1}{2} = \frac{1}{2} \].This demonstrates that the limit of the original expression as \( x \rightarrow 0 \) is indeed \( \frac{1}{2} \).

Key Concepts

Trigonometric Identities

Trigonometric identities are useful tools that allow us to transform and simplify expressions involving trigonometric functions. They rely on the fundamental properties of these functions, creating a consistent framework for various mathematical manipulations. For instance,

• The identity \( \sec \theta = \frac{1}{\cos \theta} \) helps us understand the relationship between the secant and cosine functions.
• Another key identity is \( \sec^2 \theta = 1 + \tan^2 \theta \), which connects the secant and tangent.

These identities simplify complex expressions, making calculations more manageable.
In the original exercise, the identity \( \sec^2 \frac{x}{2} = 1 + \tan^2 \frac{x}{2} \) was vital in converting the complex expression under a square root into a much more approachable form. Simplifying the expression to \( \sqrt{\tan^2 \frac{x}{2}} \) allows for easier calculations, especially when considering limits or approximations around specific values.

Derivatives

Derivatives describe how a function changes as its input changes. They are the backbone of calculus, providing insight into the rate of change and the behavior of functions.

• For instance, the derivative of \( \tan x \) is \( \sec^2 x \), showing how this trigonometric function's rate of change is tied to its value.
• Understanding derivatives helps in approximating functions around certain points, which is useful in limit calculations.

Although derivatives were not directly calculated in this exercise, the concept behind them facilitates understanding approximations and transformations occurring within the limits. The process of using a derivative approximation \( \tan \frac{x}{2} \approx \frac{x}{2} \) near zero illustrates the principle of using derivatives to predict function behavior around specific points. This approximation helps streamline the calculation of limits and uncovers hidden simplifications.

Approximation Near Zero

Approximation near zero simplifies the behavior of functions when working with limits. When \( x \) is approaching zero, trigonometric functions like sine and tangent can be approximated by simpler linear expressions.
For example:

• \( \sin x \approx x \) when \( x \) is near 0.
• Similarly, \( \tan x \approx x \) in these scenarios.

The exercise effectively uses the approximation \( \tan \frac{x}{2} \approx \frac{x}{2} \), streamlining the limit calculation. This simplifies \( \lim _{x \rightarrow 0} \frac{\left| \tan \frac{x}{2} \right|}{x} \) into an expression that is easier to handle—namely \( \lim _{x \rightarrow 0} \frac{\left| \frac{x}{2} \right|}{x} \). By canceling out similar terms and understanding behavior close to zero, we reveal the underlying outcome of such limits. Approximations near zero are crucial for determining limits effectively and efficiently.