Problem 30 - Calculus for JEE ADVANCED

Question

\(\lim _{n \rightarrow \infty}\left[\frac{2}{2-\frac{1}{n^{2}}} \cdot \frac{1}{n} \cos \left(\frac{1+1 / n}{2-1 / n}\right)-\frac{1}{\left(\frac{1}{n}-2\right)} \cdot \frac{1}{\left(1+\frac{1}{n^{2}}\right)} \cdot \frac{1}{n}\right]\)

Step-by-Step Solution

Answer

The limit is 0 as \( n \to \infty \).

Step 1: Simplifying the First Expression

Examine the first term in the limit expression: \( \frac{2}{2-\frac{1}{n^{2}}} \cdot \frac{1}{n} \cos \left(\frac{1+1 / n}{2-1 / n}\right) \). As \( n \to \infty \), the term \( \frac{1}{n} \to 0 \) and \( \cos \left(\frac{1+1 / n}{2-1 / n}\right) \approx \cos(0) = 1 \). Thus, this part simplifies to \( \frac{2}{2} \cdot \frac{1}{n} = \frac{1}{n} \).

Step 2: Simplifying the Second Expression

Consider \( \frac{1}{\left(\frac{1}{n}-2\right)} \cdot \frac{1}{\left(1+\frac{1}{n^{2}}\right)} \cdot \frac{1}{n} \). As \( n \to \infty \), \( \frac{1}{n} \approaches 0 \), \( \left(1+\frac{1}{n^{2}}\right) \approaches 1 \), and \( \frac{1}{n} - 2 \approaches -2 \). Therefore, this expression simplifies approximately to \( \frac{-1}{2n} \).

Step 3: Combining and Calculating the Limit

Now, combine the simplified expressions: \( \frac{1}{n} - \left(-\frac{1}{2n}\right) = \frac{1}{n} + \frac{1}{2n} = \frac{3}{2n} \). As \( n \to \infty \), this expression tends to zero.

Step 4: Conclusion

Since all simplified terms go to zero as \( n \to \infty \), the overall limit of the original expression is 0.

Key Concepts

Infinite Limits

Infinite limits arise when we evaluate a limit as the variable approaches infinity or negative infinity. Essentially, as the variable grows larger and larger, we try to determine what value the function approaches, if any.
In the given exercise, we are focusing on evaluating the limit as \( n \) approaches infinity. As \( n \) becomes infinitely large, certain terms of the function either diminish (such as \( \frac{1}{n} \)), signifying that they have negligible impact on the function's overall behavior, or they become constant (such as \( \cos(0) = 1 \)).
It's crucial to identify these behaviors because they allow us to simplify complex expressions. By understanding the main contributors to a function's behavior as it approaches infinity, we can determine the infinite limit of the entire expression.

Simplifying Expressions

Simplifying expressions is a critical step in solving limits and requires an understanding of how various algebraic manipulations affect the expression. This process involves breaking down complicated functions into more manageable pieces, analyzing each one separately, and identifying terms that tend toward zero or a constant as \( n \) approaches infinity.
In the original problem, the expressions contain terms like \( \frac{1}{n} \) and \( \frac{1}{n^2} \), which both tend toward zero as \( n \) becomes very large. This simplification allows us to remove negligible terms and focus on those parts of the expression that predominantly influence its behavior. For example, the cosine term simplifies greatly when considering its limit, and certain divisions simplify into constants.
By carefully simplifying each component, the overall expression becomes easier to understand and to solve. Proper notation and accuracy in these steps are crucial to arriving at the correct result.

Trigonometric Limits

Understanding trigonometric limits is essential when they appear in calculus problems, especially where trigonometric functions like sine or cosine are involved in limits as variables grow large.
In the given task, we examined \( \cos \left(\frac{1+1/n}{2-1/n}\right) \). As \( n \) approaches infinity, the expression simplifies deeply because both the numerator and denominator tend towards constants:\( \frac{1}{n} \to 0 \). This approximates the angle to zero, allowing us to conclude that \( \cos(0) = 1 \).
Typically, trigonometric limits become less complex when the arguments simplify to basic trigonometric angles such as 0 or \( \pi \), letting us utilize known values for these angles efficiently. In such cases, the focus often shifts from the trigonometric function itself to the algebraic simplification of its argument.