Problem 6 - A Complete Resource Book in Mathematics for JEE Main

Question

\(\lim _{n \rightarrow \infty} \frac{n^{k} \sin ^{2}(n !)}{n+2} 0

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Understanding the Limit

We are given the expression \( \lim _{n \rightarrow \infty} \frac{n^{k} \sin ^{2}(n !)}{n+2} \) with \( 0

Step 2: Simplifying the Sine Function

The function \( \sin(n!) \) oscillates between -1 and 1 for any integer \( n \). Therefore, \( \sin^2(n!) \) always lies between 0 and 1, no matter the value of \( n \). This implies that \( 0 \leq \sin^2(n!) \leq 1 \).

Step 3: Analyzing the Main Expression

Consider the main expression \( \frac{n^{k} \sin^2(n!)}{n+2} \). Because \( \sin^2(n!) \) is bounded between 0 and 1, the fraction is primarily affected by the term \( \frac{n^k}{n+2} \).

Step 4: Estimating the Behavior of \( \frac{n^k}{n+2} \)

For very large \( n \), the expression \( \frac{n^k}{n+2} \) behaves similarly to \( \frac{n^k}{n} = n^{k-1} \). Since \( 0 < k < 1 \), this simplifies to \( n^{k-1} \rightarrow 0 \) as \( n \rightarrow \infty \) because \( k-1 < 0 \).

Step 5: Taking the Limit

Combining the bounds on \( \sin^2(n!) \) with the behavior of \( \frac{n^k}{n+2} \), we evaluate \( \lim_{n \rightarrow \infty} \frac{n^k \sin^2(n!)}{n+2} \). Since \( \frac{n^k}{n+2} \rightarrow 0 \) and \( \sin^2(n!) \) is bounded, the entire limit tends to 0.

Key Concepts

Factorial Analysis

In this problem, we are asked to evaluate the limit involving factorials embedded within trigonometric functions. Factorial, denoted as \( n! \), is the product of all positive integers up to \( n \). It's a fundamental component in combinatorial mathematics and grows extraordinarily fast with increasing \( n \).
This rapid growth is crucial in Calculus limit problems, especially when factorials are used within oscillating functions like sine. By understanding how quickly factorials increase, we realize that they dominate the behavior of expressions they are part of.
Since \( n! \) grows so rapidly, even complicated expressions with subtle dependencies can often be simplified, especially when assessing limits and asymptotic behaviors. In this context, despite being nested in a sine function, factorials contribute importantly by highlighting how the rest of the expression is impacted by their presence. As a result, factorials can lead to rapid oscillations when used in trigonometric functions.

Sine Function Oscillation

The sine function, denoted as \( \sin(x) \), oscillates between -1 and 1. When \( x \) is a very large number like \( n! \), the oscillation still remains strictly bounded. The transformation \( \sin^2(x) \) simply translates this oscillation into the range 0 to 1, since squaring removes any negative values.
This property is utilized in our problem because it helps simplify the overall behavior: since \( \sin^2(n!) \) is always between 0 and 1, it acts like a moderating factor over the larger expression. This boundedness is key when evaluating the impact of high-growth terms like \( n^k \).
By analyzing \( \sin^2(n!) \), we essentially simplify one of the most unpredictable behaviors in the trigonometric domain. Consequently, the oscillation doesn’t affect the limit directly, but it does serve to streamline our calculations by offering a solid range within which the expression must reside.

Limit Estimation Techniques

Calculus often involves estimating complex limits to understand behavior as \( n \to \infty \). The expression \( \frac{n^k \sin^2(n!)}{n+2} \) is a typical instance, demonstrating real-world application of these techniques.
To evaluate this limit, notice that the term \( n^k \) dominates over any static upper bounds of \( \sin^2(n!) \). Yet, because \( \frac{n^k}{n+2} \) simplifies to \( n^{k-1} \) for large \( n \), and since \( k-1 < 0 \), this simplifies our calculation, leading to the expression approaching zero.
Such understanding of limits is facilitated by breaking down expressions into smaller growth terms. This process frequently involves assuming very high values whereby the relationship between variables simplifies, allowing estimation techniques to predict behavior accurately. Here, both the growth rates and mitigating factors are key, indicating the limit tends inexorably towards zero.

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Problem 7

Other exercises in this chapter

Problem 4

The value of \(\lim _{x \rightarrow \infty} x\left[\tan ^{-1} \frac{x+1}{x+2}-\frac{\pi}{4}\right]\) is (A) \(\frac{1}{2}\) (B) \(-\frac{1}{2}\) (C) 1 (D) \(-1\

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Problem 5

\(\lim _{n \rightarrow \infty} \cos \left(\pi \sqrt{n^{2}+n}\right), n \in Z\) is equal to (A) 0 (B) 1 (C) \(-1\) (D) None of these

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Problem 7

\(\lim _{x \rightarrow 1} \frac{\sqrt{1-\cos 2(x-1)}}{x-1}\) (A) exists and it equals \(\sqrt{2}\) (B) exists and it equals \(-\sqrt{2}\) (C) Does not exist bec

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Problem 8

The value of \(\lim _{x \rightarrow \infty} \frac{x^{5}}{5^{x}}\) is (A) 1 (B) \(-1\) (C) 0 (D) None of these

View solution

Key Concepts

Factorial Analysis

In this problem, we are asked to evaluate the limit involving factorials embedded within trigonometric functions. Factorial, denoted as \( n! \), is the product of all positive integers up to \( n \). It's a fundamental component in combinatorial mathematics and grows extraordinarily fast with increasing \( n \).
This rapid growth is crucial in Calculus limit problems, especially when factorials are used within oscillating functions like sine. By understanding how quickly factorials increase, we realize that they dominate the behavior of expressions they are part of.
Since \( n! \) grows so rapidly, even complicated expressions with subtle dependencies can often be simplified, especially when assessing limits and asymptotic behaviors. In this context, despite being nested in a sine function, factorials contribute importantly by highlighting how the rest of the expression is impacted by their presence. As a result, factorials can lead to rapid oscillations when used in trigonometric functions.

Sine Function Oscillation

The sine function, denoted as \( \sin(x) \), oscillates between -1 and 1. When \( x \) is a very large number like \( n! \), the oscillation still remains strictly bounded. The transformation \( \sin^2(x) \) simply translates this oscillation into the range 0 to 1, since squaring removes any negative values.
This property is utilized in our problem because it helps simplify the overall behavior: since \( \sin^2(n!) \) is always between 0 and 1, it acts like a moderating factor over the larger expression. This boundedness is key when evaluating the impact of high-growth terms like \( n^k \).
By analyzing \( \sin^2(n!) \), we essentially simplify one of the most unpredictable behaviors in the trigonometric domain. Consequently, the oscillation doesn’t affect the limit directly, but it does serve to streamline our calculations by offering a solid range within which the expression must reside.

Limit Estimation Techniques

Calculus often involves estimating complex limits to understand behavior as \( n \to \infty \). The expression \( \frac{n^k \sin^2(n!)}{n+2} \) is a typical instance, demonstrating real-world application of these techniques.
To evaluate this limit, notice that the term \( n^k \) dominates over any static upper bounds of \( \sin^2(n!) \). Yet, because \( \frac{n^k}{n+2} \) simplifies to \( n^{k-1} \) for large \( n \), and since \( k-1 < 0 \), this simplifies our calculation, leading to the expression approaching zero.
Such understanding of limits is facilitated by breaking down expressions into smaller growth terms. This process frequently involves assuming very high values whereby the relationship between variables simplifies, allowing estimation techniques to predict behavior accurately. Here, both the growth rates and mitigating factors are key, indicating the limit tends inexorably towards zero.