Problem 14 - Calculus for Biology and Medicine

Question

Evaluate the trigonometric limits. $$ \lim _{x \rightarrow 0} \frac{1-\cos (x / 2)}{x} $$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Recognize trigonometric identity

We will use the trigonometric identity \( 1 - \cos A = 2 \sin^2(\frac{A}{2}) \) to simplify the expression. For our limit, substitute \( A = \frac{x}{2} \), giving us \( 1 - \cos(\frac{x}{2}) = 2 \sin^2(\frac{x}{4}) \).

Step 2: Substitute the identity

Replace \( 1 - \cos(\frac{x}{2}) \) with \( 2\sin^2(\frac{x}{4}) \) inside the limit. The expression becomes: \[ \lim _{x \rightarrow 0} \frac{2 \sin^2(\frac{x}{4})}{x} \]

Step 3: Simplify the expression

Factor out and rearrange the expression: \[ \lim _{x \rightarrow 0} \frac{2 \sin^2(\frac{x}{4})}{x} = 2 \lim _{x \rightarrow 0} \frac{\sin^2(\frac{x}{4})}{x} \] Now introduce quadratic expression, \( \sin^2(\frac{x}{4}) = (\sin(\frac{x}{4}))^2\), which allows further simplification.

Step 4: Apply double angle formulas

We use the fact that \[ \sin(\frac{x}{4}) \approx \frac{x}{4} \text{ as } x \to 0 \]Consequently, \[ (\sin(\frac{x}{4}))^2 \approx (\frac{x}{4})^2 \]. Substitute this into the expression:\[ 2 \lim _{x \rightarrow 0} \frac{(\frac{x}{4})^2}{x} = 2 \lim _{x \rightarrow 0} \frac{x^2}{16x} = 2 \lim _{x \rightarrow 0} \frac{x}{16} \].

Step 5: Evaluate the limit

The simplified expression \[ 2 \lim _{x \rightarrow 0} \frac{x}{16} = 2 \cdot \frac{0}{16} = 0 \]. Hence, the original limit evaluates to zero.

Key Concepts

Trigonometric Identities

Trigonometric identities are valuable tools in calculus and algebra. They allow us to simplify and manipulate trigonometric expressions, making complex problems easier to manage. In this exercise, the identity \(1 - \cos A = 2 \sin^2(\frac{A}{2})\) is particularly useful. This identity can be quite handy when faced with limits involving cosine functions that are hard to evaluate directly.
For example, to solve the limit \(\lim _{x \rightarrow 0} \frac{1-\cos (x / 2)}{x}\), we substitute \(A = \frac{x}{2}\). This substitution changes our problem to \(1 - \cos(\frac{x}{2}) = 2 \sin^2(\frac{x}{4})\), transforming the original expression.

• This shift in focus allows us to work with sine squared terms instead of cosine, which often simplifies the evaluation of limits.
• Recognizing and applying these identities can drastically reduce the complexity of trigonometric limit problems.

Limit Evaluation

Evaluating limits often serves as a critical step when analyzing functions in calculus. It's about understanding how a function behaves as the input approaches a particular value.
In our example, we start with \(\lim _{x \rightarrow 0} \frac{2 \sin^2(\frac{x}{4})}{x}\). The challenge is to simplify this expression as \(x\) approaches 0. By first substituting the trigonometric identity, the rewritten expression \(2 \lim _{x \rightarrow 0} \frac{\sin^2(\frac{x}{4})}{x}\) can be further broken down.

• The sine function approximation \(\sin(\frac{x}{4}) \approx \frac{x}{4}\) for small values of \(x\) narrows down the expression to a manageable form.
• The quadratic form \((\sin(\frac{x}{4}))^2 \approx (\frac{x}{4})^2\) further aids in simplifying.

These approximations reveal the behavior close to \(x = 0\), allowing the limit \(2 \lim _{x \rightarrow 0} \frac{(\frac{x}{4})^2}{x}\) to eventually resolve to 0.

Calculus Techniques

Calculus techniques are indispensable when solving limits involving trigonometric functions. A common technique involves substitution, as seen with the triadic change from cosine to sine squared terms. This was our first step towards simplification.
Then, we use small-angle approximations like \(\sin(\frac{x}{4}) \approx \frac{x}{4}\). This approximation becomes valuable as \(x\) approaches zero, transforming the complex trigonometric expressions into simpler polynomial forms.

• Factoring out constants and simplifying the equation makes expressions manageable and limits easier to evaluate.
• Being familiar with common trigonometric approximations and identities is crucial for tackling such problems.
• Finally, understanding the linear behavior of functions around points, especially as \(x\) approaches zero, consolidates the calculus approach, providing neat and simplified results.

These techniques, while basic, form the backbone of more advanced calculus, demonstrating the beauty of simplification in problem-solving.