Problem 47 - Calculus for JEE ADVANCED

Question

\(\lim _{x \rightarrow x} \frac{1}{x+1} \tan \left(\frac{\pi x+1}{2 x+2}\right)=\lim _{x \rightarrow \infty} \frac{1}{x+1} \cot \left(\frac{\pi}{2}-\frac{\pi x+1}{2 x+2}\right)\)

Step-by-Step Solution

Answer

Both limits are equal to 0.

Step 1: Understanding the problem

We need to evaluate both limits as \( x \) approaches infinity and determine whether they are equal. The functions supplied involve trigonometric identities that suggest transformations can simplify them.

Step 2: Simplify the trigonometric expression

First, consider \( \tan\left(\frac{\pi x+1}{2x+2}\right) \). To simplify \( \cot\left(\frac{\pi}{2} - \frac{\pi x+1}{2x+2}\right) \), use the identity \( \cot(\alpha) = \tan(\frac{\pi}{2} - \alpha) \), thus it is transformed into \( \tan\left(\frac{\pi x+1}{2x+2}\right) \).

Step 3: Equivalence realization

Both expressions \( \tan\left(\frac{\pi x+1}{2x+2}\right) \) and \( \cot\left(\frac{\pi}{2} - \frac{(\pi x+1)}{2x+2}\right) \) simplify into the same form. This occurs due to the identity transformation in the \( \cot \) function.

Step 4: Consider limit properties

As \( x \to \infty \), observe \( \frac{1}{x+1} \). Since \( \frac{1}{x+1} \to 0 \), the influence of the term \( \tan\left(\frac{\pi x+1}{2x+2}\right) \) multiplied by it also approaches 0 due to the bounded nature of the trigonometric function.

Step 5: Apply the limit

Evaluate \( \lim_{x \to \infty} \frac{1}{x+1} \tan\left(\frac{\pi x+1}{2x+2}\right) \). This simplifies to \( 0 \) as \( \frac{1}{x+1} \) approaches 0 and \( \tan\left(\frac{\pi x+1}{2x+2}\right) \) remains finite.

Step 6: Verification by symmetry

Verify that \( \lim_{x \to \infty} \frac{1}{x+1} \cot\left(\frac{\pi}{2} - \frac{\pi x+1}{2x+2}\right) \) similarly approaches \( 0 \). Since both original expressions simplify into the same function, the limits are consequently equal.

Key Concepts

Trigonometric Identities

Trigonometric identities are like shortcuts for simplifying trigonometric expressions. Often, they can turn a complex problem into something more manageable. Here's how they work:

• They define relationships between different trigonometric functions like sine, cosine, tangent, and cotangent.
• One of the most useful identities is the Pythagorean identity: \( \sin^2(\theta) + \cos^2(\theta) = 1 \).
• Another important identity used in this problem is the complementary angle identity, which tells us that \( \cot(\alpha) = \tan\left(\frac{\pi}{2} - \alpha\right) \).

Whenever you encounter a trigonometric function, check if an identity can simplify your expression.
In the original problem, the expression \( \cot\left(\frac{\pi}{2} - \frac{\pi x+1}{2x+2}\right) \) simplifies using the complementary angle identity to \( \tan\left(\frac{\pi x+1}{2x+2}\right) \).Identities like these are incredibly powerful in calculus, especially when evaluating limits.

Infinite Limits

Infinite limits are all about understanding how a function behaves as its input grows infinitely large or small. Understanding infinite limits can help you predict what a function will do even at extreme values. Here are the main ideas:

• An infinite limit occurs when the value of a function increases or decreases without bound as the variable approaches a specific point.
• Alternatively, it can mean that the function approaches a particular value as the input reaches infinity.
• Consider functions involving fractions: as the denominator becomes infinitely large, the fraction's overall value tends towards zero.

In our exercise, when \( x \) becomes large (approaches infinity), \( \frac{1}{x+1} \) becomes very small, effectively tending towards zero. This is because any number over an increasingly large denominator will tend to be close to zero.Thus, as \( x \to \infty \), the original function involving \( \frac{1}{x+1} \tan\left(\frac{\pi x+1}{2x+2}\right) \) approaches zero, owing to the limit properties of the terms involved.

Function Transformations

Function transformations help us modify and manipulate functions to study their properties or simplify them. These transformations can occur in various forms:

• Translation: Moving the graph of the function up, down, left, or right without changing its shape.
• Scaling: Stretching or compressing the graph vertically or horizontally.
• Reflection: Flipping the graph over a specific axis.

In the exercise, transformations involving trigonometric functional modifications show us how the trigonometric expression was simplified.Using these transformations, we simplified \( \cot\left(\frac{\pi}{2} - \frac{\pi x+1}{2x+2}\right) \) to \( \tan\left(\frac{\pi x+1}{2x+2}\right) \), relying on trigonometric identities to recognize that both resemble each other closely, simplifying further evaluation. Understanding how transformations work empowers you to manipulate and simplify complex functions, making calculus problems much more approachable.