Problem 2 - Calculus for Biology and Medicine

Question

Evaluate the trigonometric limits. $$ \lim _{x \rightarrow 0} \frac{\sin (2 x)}{4 x} $$

Step-by-Step Solution

Answer

The limit is \( \frac{1}{4} \).

Step 1: Identify the Basic Limit

Recognize that there is a basic trigonometric limit commonly used for evaluating this type of problem: \( \lim_{x \rightarrow 0} \frac{\sin x}{x} = 1 \). We can apply this by rewriting the limit in a similar form.

Step 2: Rewrite the Expression

Rewrite the original expression \( \frac{\sin (2x)}{4x} \) in terms of the basic limit. Notice that we can express it as \( \frac{1}{4} \times \frac{\sin (2x)}{2x} \). This allows us to use the basic limit by considering \( 2x \) as the variable approaching zero.

Step 3: Apply the Limit

Apply the basic limit to \( \frac{\sin (2x)}{2x} \) by recognizing that as \( x \rightarrow 0 \), \( 2x \rightarrow 0 \) as well, hence \( \lim_{x \rightarrow 0} \frac{\sin (2x)}{2x} = 1 \).

Step 4: Multiply by Constant

Combine the findings by multiplying the limit found in Step 3 by the constant that was factored out in Step 2: \( \frac{1}{4} \times 1 = \frac{1}{4} \).

Key Concepts

Understanding Basic Trigonometric Limits

One of the core concepts in calculus is understanding how to evaluate basic trigonometric limits. These are fundamental in dealing with problems involving trigonometric functions, especially as they approach a particular point, often zero. The most common form you will encounter is the limit \( \lim_{x \rightarrow 0} \frac{\sin x}{x} = 1 \). This expression holds a special place in calculus as it provides a foundation for evaluating more complex trigonometric expressions at their limits.A key aspect of using this limit is transforming your trigonometric expression to closely resemble this basic form. For instance, if you have a limit such as \( \frac{\sin (2x)}{x} \), you will want to manipulate it to look like \( \frac{\sin x}{x} \) by altering coefficients or denominators to isolate the \( \sin \, \text{angle} \) and corresponding 'angle' in the denominator. This manipulation allows you to directly apply the basic limit efficiently.

Approach to Limit Evaluation

When evaluating limits, particularly trigonometric ones, the process often involves rewriting the expression to reveal any hidden forms of these well-known limits. Start by inspecting the expression to spot patterns or factors that resemble basic limits. - For instance, with \( \lim _{x \rightarrow 0} \frac{\sin (2 x)}{4 x} \), recognize it needs adjustment to apply the basic limit. - First, rewrite it in the form \( \frac{\sin (2x)}{2x} \) which is possible by factoring. - You express it as \( \frac{1}{4} \times \frac{\sin (2x)}{2x} \).Once you have isolated the trigonometric part in a form where it can leverage the basic limit theorem, you can solve the limit. Use the known result that \( \lim_{x \rightarrow 0} \frac{\sin x}{x} = 1 \) for simplification, which will lead directly to your answer.This method not only simplifies your calculations but also solidifies your understanding of limit properties and their applications.

Effective Calculus Problem Solving

Problem-solving in calculus involves strategic thinking and an understanding of key mathematical rules and theorems. Especially in trigonometric limits, the objective is to transform the given problem into something familiar, such as isolated basic trigonometric limits.Here are general tips for effective calculus problem solving:- Break down the problem into manageable pieces; identify constants and parameters that can be factored out.- Use simplification techniques to transform expressions; for trigonometric limits, this often involves rewriting the trigonometric function and its angle in the denominator.- Factor, reduce, and isolate terms when possible to reveal applicable limit forms.- Calculate carefully, ensuring each step logically follows from the previous.Applying these strategies to a problem like \( \lim _{x \rightarrow 0} \frac{\sin (2 x)}{4 x} \) helps you not only find the answer, which is \( \frac{1}{4} \), but also gain a deeper understanding of the application of calculus in trigonometry. This approach fosters a robust problem-solving mindset invaluable for tackling complex calculus problems.