Problem 90 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The value of \(\lim _{x \rightarrow \infty}|x|^{[\cos x]}\), where \([\cdot]\) denotes the greatest integer, is (A) 0 (B) 1 (C) \(-1\) (D) Does not exist

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Understanding the Function

We need to evaluate the limit \( \lim_{x \to \infty} |x|^{[\cos x]} \). The expression involves two parts: \( |x| \), which is the absolute value of \( x \), and \( [\cos x] \), which is the greatest integer function applied to \( \cos x \).

Step 2: Analyzing \( \cos x \)

As \( x \) approaches infinity, \( \cos x \) will oscillate between -1 and 1. Therefore, the value of the greatest integer function \( [\cos x] \) is dependent on the value of \( \cos x \). Particularly, \( [\cos x] = 0 \) when \( -1 < \cos x < 1 \), which holds true for most \( x \) as it cycles between -1 and 1.

Step 3: Simplifying the Expression

For the vast majority of \( x \), given that \( [\cos x] = 0 \), the original expression becomes \( |x|^0 \), which simplifies to 1, since any non-zero number raised to the power 0 is 1.

Step 4: Evaluating the Limit

Since \( [\cos x] = 0 \) for the vast majority of values \( x \) as it approaches infinity, the expression \( |x|^0 = 1 \) will hold for most of these values. Thus, the limit of this expression as \( x \to \infty \) is clearly 1.

Key Concepts

Greatest Integer Function

The greatest integer function, often denoted as \([\cdot]\), is quite an interesting mathematical concept. It takes a real number and rounds it down to the nearest integer. Essentially, it "chops off" the decimal part of a number, leaving you with an integer. For example:

• If you have 3.7, the greatest integer function, \([3.7]\), gives you 3.
• For -1.2, the function \([-1.2]\) results in -2.

This function is particularly useful in situations where approximation to the nearest whole number is required. In the exercise, \([\cos x]\) was part of the expression. Here, the value of \([\cos x]\) is crucial because \[\cos x\] affects the exponent to which the absolute value of \(x\), \(|x|\), is raised.

Oscillating Functions

Oscillating functions, such as \(\cos x\), are those that regularly vary between two limits. For \(\cos x\), these limits are -1 and 1, and the function cycles through these values as \(x\) increases. The periodic nature of \(\cos x\) means it does not settle on a single value, but continually moves in a wave-like fashion.
This oscillatory behavior is important when considering limits. As the exercise shows, because \(\cos x\) never stays fixed, the greatest integer value \([\cos x]\) can change. However, it mostly remains at 0 due to the fact that \(\cos x\) is rarely exactly -1 or 1 for an infinitesimal point in its cycle. Consequently, this causes the exponent in \(|x|^{[\cos x]}\) to often simplify to 0, impacting the final result significantly.

Exponential Functions

Exponential functions involve numbers raised to the power of another value, forming expressions like \(a^b\). This type of function can grow rapidly or decay, depending on the exponent. In the exercise, you encountered \(|x|^{[\cos x]}\). Here, \(a = |x|\) and \(b = [\cos x]\).
The fascinating part about exponential functions is their sensitivity to changes in the exponent. Small differences in the exponent can lead to vastly different results.

• When \(b = 0\), any number except zero raised to this power becomes 1 (e.g., 5 exttt{⁰} = 1).
• When \(b = 1\), the result is simply the base itself.
• If \(b\) is negative, the function describes exponential decay.

In the limit problem, most often \(b = [\cos x] = 0\), simplifying the expression to \(|x|^0 = 1\). This simplification is a key step in determining the limit as \(x\) approaches infinity.