Problem 22 - Calculus 1 Course in Mathematics for the IIT JEE and Other Engineering Exams

Question

\(\lim _{x \rightarrow 0} \frac{\sin \left(\pi \cos ^{2} x\right)}{x^{2}}=\) (a) \(-\pi\) (b) \(\pi\) (c) \(\pi / 2\) (d) 1

Step-by-Step Solution

Answer

Limit is 0

Step 1: Understand the Problem

We need to find the limit of the expression \(\lim _{x \rightarrow 0} \frac{\sin \left(\pi \cos ^{2} x\right)}{x^{2}}\). It involves the sine function and a limit approaching zero, which suggests using series expansion or known limit results.

Step 2: Simplify \(\cos^2 x\) at \(x \to 0\)

Recall that \(\cos x\) can be expressed as \(1 - \frac{x^2}{2} + \text{higher order terms}\). Hence, \(\cos^2 x = \left(1 - \frac{x^2}{2}\right)^2 \approx 1 - x^2 + \frac{x^4}{4}\).

Key Concepts

Trigonometric Limits

Trigonometric limits are a common occurrence in calculus, especially when we deal with functions involving sine, cosine, and tangent as variables approach zero or infinity. One of the foundational results is the limit involving sine:

• As \( x \to 0 \), \( \frac{\sin x}{x} \to 1 \).

This particular case is useful because expressions involving sine or cosine functions approaching zero can often be simplified using this result. Additionally, keeping in mind the small angle approximations like \( \sin \theta \approx \theta \) and \( \cos \theta \approx 1 - \frac{\theta^2}{2} \) is crucial, as these can greatly simplify complex trigonometric expressions when limits are involved. Understanding these limits is vital for solving problems that include squeezing techniques and problems where the function is not easily differentiated.

Series Expansion

Series expansion is a powerful tool in calculus that simplifies functions into polynomials by breaking them down into their component parts. For instance, the Taylor or Maclaurin series are often employed for this purpose, leading to easier handling of limits or derivatives. When solving complex limit problems, especially those involving trigonometric functions, it's particularly useful to express the functions as a series. In our exercise, for example, the function \( \cos x \) around \( x = 0 \) can be expressed as

• \( \cos x = 1 - \frac{x^2}{2} + \frac{x^4}{24} + \cdots \)

For \( \cos^2 x \), this results in successive simplification, making it easier to assess the behavior of the function as \( x \to 0 \). This simplification reduces the need to compute exact trigonometric values, making limits easier to find by focusing just on a few primary terms.

L'Hôpital's Rule

L'Hôpital's Rule provides a technique to evaluate limits of indeterminate forms \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). It allows us to take derivatives of the numerator and the denominator separately until we can evaluate the limit:

• If \( \lim_{x \to a} f(x) = 0 \) and \( \lim_{x \to a} g(x) = 0 \), then \( \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} \), provided this limit exists.

In our exercise, if the expression allows for L'Hôpital's rule, we could differentiate the sine and the polynomial in the denominator, resulting in a simpler expression for finding the limit. However, in many instances, the problem can be tackled using series expansions or known limit results, as differentiation might not always simplify the expression adequately.

Mathematical Problem Solving

Problem solving in mathematics involves understanding the problem, devising a plan, carrying out that plan, and then reassessing the result. With calculus and limits, this often means recognizing patterns or forms that are already familiar and seeing how rules can be applied:

• Step One: Translate the problem into mathematical language and identify known forms or limits that might be useful.
• Step Two: Use simplifications such as series expansions to transform complex expressions into more manageable forms.
• Step Three: Apply appropriate rules, like L'Hôpital's Rule, when you encounter indeterminate forms.
• Step Four: Verify the solution by confirming that all approximations are valid and that the original problem constraints are satisfied.

This structured approach not only aids in solving the problem but also enhances understanding, ensuring that the solution is not just a conclusion but a comprehensive learning experience.