Problem 18 - 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 Standard Limit

You need to evaluate the limit \( \lim _{x \rightarrow 0} \frac{1-\cos (x / 2)}{x} \). Recall the standard trigonometric limit: \( \lim_{u \to 0} \frac{1-\cos(u)}{u^2} = \frac{1}{2} \). To use this, we need to rewrite the expression to match this form.

Step 2: Substitute Variable

Set \( u = \frac{x}{2} \) so that when \( x \to 0 \), \( u \to 0 \) as well. Note that \( x = 2u \), so rewrite the limit in terms of \( u \): \( \lim_{u \to 0} \frac{1 - \cos(u)}{2u} \).

Step 3: Factor Out Scalar

Take \( \frac{1}{2} \) outside the limit since it is a constant multiplier: \[ \frac{1}{2} \lim_{u \to 0} \frac{1 - \cos(u)}{u} \].

Step 4: Use Trigonometric Identity

Apply the trigonometric limit \( \lim_{u \to 0} \frac{1-\cos(u)}{u^2} = \frac{1}{2} \). Here, we need \( u \) in the denominator squared. Rewrite the limit expression as \( \frac{1}{2} \cdot \lim_{u \to 0} \left( \frac{1 - \cos(u)}{u^2} \cdot u \right) \).

Step 5: Evaluate the Limit

Using the trigonometric limit identity: \( \lim_{u \to 0} \frac{1-\cos(u)}{u^2} = \frac{1}{2} \), therefore, \( \frac{1}{2} \lim_{u \to 0} \frac{1 - \cos(u)}{u^2} \cdot u = \frac{1}{2} \cdot \frac{u}{u^2} = \frac{1}{2} \cdot \frac{1}{u} \). As \( u \to 0 \), this gives \( 0 \).

Step 6: Conclusion

Combine the results of the previous steps to conclude that: \( \lim _{x \rightarrow 0} \frac{1-\cos (x / 2)}{x} = 0 \).

Key Concepts

Standard Trigonometric Limit

One of the essential tools for evaluating limits involving trigonometric functions is the standard trigonometric limit. This is particularly useful for expressions that tend to a limit as the variable approaches zero. The key standard limit in this context is:

• \( \lim_{u \to 0} \frac{1 - \cos(u)}{u^2} = \frac{1}{2} \).

These limits help simplify complex trigonometric limit problems by transforming them into known forms that are easier to evaluate. This particular limit is handy in situations where the cosine function is involved and the expression consists of differences from one.
Recognizing the form of this standard limit is often the first step in solving the given problem.

Limit Evaluation

Evaluating the limit requires understanding how the function behaves as the variable approaches a specific value, in this case, zero. In our problem, we need to evaluate:\( \lim _{x \rightarrow 0} \frac{1-\cos (x / 2)}{x} \).This involves rewriting or manipulating the expression into a form that reflects standard limits. Here, recognizing the '1 - cos(u)' pattern was crucial, so we could use it appropriately. By utilizing known limits, we can more easily determine the behavior of complex trigonometric expressions as they approach specific values.
Both algebraic manipulation and knowledge of trigonometric characteristics are essential for evaluating such limits effectively.

Substitution Method

The substitution method is a powerful technique for simplifying limit problems. In this exercise, we used the substitution:\( u = \frac{x}{2} \).This substitution gives us several benefits:

• It transforms the variable \( x \) into a simpler form \( u \), which handles half-angles effectively.
• It simplifies the limit expression into \( \lim_{u \to 0} \frac{1 - \cos(u)}{2u} \).
• It allows us to apply standard limit evaluations conveniently.

Substitution is particularly useful when direct computation of the limit becomes cumbersome or when standard limits need to be applied to equivalent functions.

Trigonometric Identity

Using trigonometric identities can greatly simplify the process of evaluating limits. In our solution, the expression was rewritten using the identity that relates the square of the denominator:\( \frac{1}{2} \cdot \lim_{u \to 0} \left( \frac{1 - \cos(u)}{u^2} \cdot u \right) \).Understanding and applying identities such as these are crucial in trigonometry because they transform complicated expressions into more manageable ones. In this case, the trigonometric identity allowed us to apply the standard limit directly and complete the limit evaluation effectively.
In sum, these identities bridge the gap between abstract trigonometric concepts and practical solutions, simplifying the process of limit evaluation.