Problem 5 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The limit is \( \pi \).

Step 1: Apply Trigonometric Limit Property

Recall the standard trigonometric limit: \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \). To use this, we need the limit in the form \( \lim_{u \to 0} \frac{\sin u}{u} \).

Step 2: Substitute to Match the Trigonometric Limit

Let \( u = \pi x \). Thus, as \( x \to 0 \), \( u \to 0 \) as well. So, \( \frac{\sin(\pi x)}{x} = \frac{\sin(u)}{u/\pi} = \pi \cdot \frac{\sin(u)}{u} \).

Step 3: Evaluate the Limit

Using the trigonometric limit \( \lim_{u \to 0} \frac{\sin u}{u} = 1 \), we have: \[\lim_{x \to 0} \frac{\sin(\pi x)}{x} = \lim_{u \to 0} \pi \cdot \frac{\sin u}{u} = \pi \cdot 1 = \pi.\]

Key Concepts

Limit Evaluation

The concept of limit evaluation forms the basis of understanding how functions behave as they approach certain points. In this particular problem, we are evaluating limits that involve trigonometric functions. The key approach involves substituting variables in such a way that they match standard limit forms. By applying known limit properties, we simplify the trigonometric expressions.

The standard trigonometric limit that many calculus students encounter is:

• \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \)

In our problem, we adjust the variable to fit this model by substituting \( u = \pi x \), which keeps the limit in a recognizable form. By substituting back and adjusting accordingly, you show that \( \lim_{x \to 0} \frac{\sin(\pi x)}{x} = \pi \). Understanding this process allows for easier manipulation and calculation of limits involving more complex functions. Deploying the method demonstrated above efficiently transforms complex situations into solvable forms.

Trigonometric Functions

Trigonometric functions such as \( \sin(x) \) play a crucial role in calculus, especially in limit evaluations and integration. Functions like sine and cosine arise frequently in calculus problems. They allow us to describe periodic and oscillatory behavior in mathematical terms.

In our given exercise, the function \( \sin(\pi x) \) is central to solving the limit problem. The sine function is an odd function with a range of \([-1, 1]\) and it is important to recognize how this behavior affects calculations with limits. By understanding properties of \( \sin(x) \), we interpret complex expressions and transform them into simpler forms.

• These transformations often involve regrouping and substituting angles or variables, such as switching from \( \sin(\pi x) \) to \( \sin(u) \).

Understanding how trigonometric functions interact with limits provides a strong foundation for tackling more advanced calculus topics.

Calculus Problem-Solving

Effective calculus problem-solving requires recognizing patterns and applying known rules efficiently. This skill is particularly necessary when dealing with limits and trigonometric functions, which can otherwise appear daunting.

To solve these types of calculus problems, begin by identifying if the limit can be transformed into a known form. Focusing on pattern recognition allows you to spot these transformations.

• In this exercise, by acknowledging the form of \( \frac{\sin x}{x} \), we quickly apply known limits.
• Using substitution techniques, such as setting \( u = \pi x \), helps align the problem with recognizable limit forms.

Committing standard limit forms and their solutions to memory is also crucial. This enables you to work through limits methodically, rapidly finding the value without unnecessary computational steps. With these skills refined, solving calculus problems becomes a matter of applying learned methods rather than reinventing solutions each time a new problem arises.