Problem 368 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \left.\lim _{\varphi \rightarrow a}\left(a^{2}-\varphi^{2}\right) \tan \frac{\pi \varphi}{2 a} \text { \\{Ans. } \frac{4 a^{2}}{\pi}\right\\} $$

Step-by-Step Solution

Answer

The limit of the function as \(\varphi\) approaches \(a\) is \(-\frac{a^{2}}{2}\)

Step 1: Identify Problem Form

When \(\varphi\) approaches the value of \(a\), the limit of the function \( \left(a^{2}-\varphi^{2}\right) \tan \frac{\pi \varphi}{2 a}\) appears to be in the form \(\frac{0}{0}\). This is because \(\left(a^{2}-\varphi^{2}\right)\) becomes \(0\) as \(\varphi\) approaches \(a\), and \(\tan \frac{\pi \varphi}{2 a}\) becomes \(\tan \frac{\pi a}{2 a} = \tan \frac{\pi}{2}\), which is undefined.

Step 2: Prepare for L'Hopital's Rule

In order to apply L'Hopital's rule, which states that for any function with a format of \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), the limit as x approaches a is the same as taking the derivative of the numerator and denominator separately and finding the limit. Rewrite the function by using the identity \(a^{2}-\varphi^{2} = (a - \varphi)(a + \varphi)\), and then divide it by \(\frac{2}{\frac{\pi \varphi}{2 a}} = \frac{4a}{\pi \varphi}\), so the function is now \(\frac{(a - \varphi)(a + \varphi) }{\frac{4a}{\pi \varphi}}\).

Step 3: Apply L'Hopital's Rule

Differentiate the numerator to become \( 2\varphi\), and the denominator to become \(-\frac{4a}{\varphi^{2}}\). The new function is now in form of \(\frac{2\varphi}{-\frac{4a}{\varphi^{2}}}\). Now if we simplify the above expression it gives \(-\frac{2\varphi^{3}}{4a} = -\frac{\varphi^{3}}{2a}\)

Step 4: Evaluate Limit

Evaluate \(-\frac{\varphi^{3}}{2a}\) as \(\varphi\) approaches \(a\), which equals to \(-\frac{a^{3}}{2a} = -\frac{a^{2}}{2}\)

Key Concepts

Limits and Continuity

Limits are foundational in calculus as they help define the behavior of functions as they approach a specific input. In this exercise, the limit deals with the function \( \left(a^{2}-\varphi^{2}\right) \tan \frac{\pi \varphi}{2 a} \). As \( \varphi \) approaches \( a \), one part of this expression, \( a^2 - \varphi^2 \), becomes 0, suggesting an indeterminate form \( \frac{0}{0} \). The concept of continuity is about smoothness of the function. Here, continuity is interrupted at \( \varphi = a \) due to the undefined nature of \( \tan \frac{\pi \varphi}{2 a} \) as it tends to infinity. For such problems, L'Hôpital's Rule is useful to address these undefined or infinite discontinuities, ensuring proper calculation of limits.

Trigonometric Limits

Trigonometric functions often present challenges in calculus, especially when considering limits. The term \( \tan \frac{\pi \varphi}{2 a} \) becomes undefined as \( \varphi \) approaches \( a \) since it heads towards \( \tan \frac{\pi}{2} \), which is not finite. This characteristic of trigonometric functions, where they don't have a defined limit at specific points, requires us to handle calculations more carefully. By understanding identities such as \( a^2 - \varphi^2 = (a - \varphi)(a + \varphi) \), we can attempt to simplify and manage these forms. Fortunately, L'Hôpital's Rule assists us by allowing us to differentiate the original function parts and find a manageable limit solution even in these tricky trigonometric scenarios.

Derivatives in Calculus

Derivatives are crucial when dealing with changing rates or slopes of curves. In this problem, the application of L'Hôpital's Rule requires differentiating the numerator and the denominator separately to resolve the indeterminate form \( \frac{0}{0} \). By taking the derivative of \( (a - \varphi)(a + \varphi) \), which simplifies to \( 2\varphi \), and differentiating the denominator, we manage to transform the expression into a solvable limit. After performing these derivations, the expression simplifies further, aiding in calculating the desired limit. This process highlights the power of derivatives as tools in calculus to tackle otherwise complex and undefined scenarios.