Problem 19 - Calculus 1 Course in Mathematics for the IIT JEE and Other Engineering Exams

Question

\(\lim _{x \rightarrow 0} \frac{1-\cos (1-\cos x)}{x^{4}}=\) (a) \(1 / 8\) (b) \(1 / 2\) (c) \(1 / 4\) (d) None of these

Step-by-Step Solution

Answer

The limit is \( \frac{1}{8} \), so the answer is (a).

Step 1: Analyze the Function

We need to find \( \lim_{x \rightarrow 0} \frac{1-\cos(1-\cos x)}{x^4} \). First, observe what happens when \( x \to 0 \). The function becomes \( \frac{1}{x^4} (1 - \cos(1-\cos 0)) = \frac{1 - \cos(0)}{x^4} = 0 \) at \( x = 0 \), which suggests the need for deeper analysis using Taylor expansions.

Step 2: Expand \( \cos x \) and \( 1 - \cos x \) Using Taylor Series

Around \( x = 0 \), the Taylor series for \( \cos x \) is approximately \( 1 - \frac{x^2}{2} + \frac{x^4}{24} + O(x^6) \). Thus, \( 1 - \cos x \approx \frac{x^2}{2} - \frac{x^4}{24} \).

Step 3: Expand \( \cos(1-\cos x) \) Using Taylor Series

Now apply the expansion to \( \cos(1-\cos x) \) using its Taylor expansion: \( \cos(1-y)\approx \cos 1 + y \sin 1 - \frac{y^2}{2} \cos 1 + O(y^3) \). Substitute \( y = \frac{x^2}{2} \), to get \[\cos(1-\frac{x^2}{2}) \approx \cos 1 - \frac{x^2}{2} \sin 1 - \frac{(\frac{x^2}{2})^2}{2} \cos 1.\]

Step 4: Simplify the Expression

Substituting these approximations back into the original limit, we have \( 1 - \cos(1-\cos x) \approximately \frac{x^4}{8} \) due to dominant terms. Thus, the expression for the limit becomes \( \lim_{x \rightarrow 0} \frac{\frac{x^4}{8}}{x^4} = \frac{1}{8} \).

Step 5: Conclude the Limit

Since the expansion simplifies the expression in the numerator to dominate as \( \frac{x^4}{8} \), the entire limit evaluates to \( \frac{1}{8} \). Therefore, the answer is (a) \( \frac{1}{8} \).

Key Concepts

Taylor series

The Taylor series is a powerful tool for approximating functions using polynomials. Imagine you need to estimate the behavior of a function around a specific point, like zero, with increasing precision. This is where the Taylor series comes in handy. It expresses a function as an infinite sum of terms. Each term is a derivative of the function at that point, multiplied by a factor involving the independent variable (like \( x \) in our example) raised to a power.

For example, the Taylor series expansion of the cosine function around \( x = 0 \) is:

• \( \cos x \approx 1 - \frac{x^2}{2} + \frac{x^4}{24} + O(x^6) \)

This representation allows you to approximate \( \cos x \) by a polynomial, which makes it easier to handle mathematically, especially when calculating limits or derivatives.

trigonometric limits

Trigonometric limits often involve functions like sine, cosine, and tangent as \( x \) approaches a particular value, often zero. These limits are crucial in calculus, especially when evaluating expressions that appear undefined or indeterminate at first sight.

In our exercise, the expression involves \( 1 - \cos(1-\cos x) \). The goal is to analyze it as \( x \) approaches zero. By using Taylor expansions for the inner cosine function, you can rewrite the expression in a simplified manner, making it easier to evaluate the limit.

Such techniques help you simplify the function into a form that reveals its behavior without the complexities of trigonometric functions.

• This is especially helpful when dealing with limits that seem tricky or cumbersome due to the presence of trigonometric functions.

cosine function

The cosine function is one of the foundational trigonometric functions in mathematics, represented by \( \cos x \). It describes the ratio of the adjacent side to the hypotenuse of a right triangle, considering an angle \( x \) measured in radians. It's an even function, meaning \( \cos(-x) = \cos(x) \).

What's fascinating about cosine is its periodic nature, repeating its values in regular intervals of \( 2\pi \). Around zero, as seen in the Taylor series, \( \cos x \) shows that it closely approximates 1. This approximation simplifies expressions where \( x \) is small, as in the original exercise's limit problem.

The function behaves predictably, making it manageable within the confines of calculus, particularly in Taylor expansions and analyzing limits.

limit analysis

Limit analysis in calculus helps you understand the behavior of expressions as a variable approaches a particular point. Expressions like \( \lim_{x \rightarrow 0} \frac{1-\cos(1-\cos x)}{x^4} \) can initially seem indeterminate. Through tools like Taylor series and simplifications, you can reveal and resolve such indeterminate forms.

For this problem, expanding \( \cos x \) and subsequently \( \cos(1-\cos x) \) into simpler polynomial forms allows you to manage the expression better. By substituting these approximations, the original denominator \( x^4 \) cancels out, letting you determine the limit value more clearly.

• This kind of manipulation is essential for transitioning from seemingly intractable expressions to solvable forms.
• In this particular case, the expanded form leads to the simple limit result of \( \frac{1}{8} \).