Problem 30 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}\) be a positive increasing function with \(\lim _{x \rightarrow \infty} \frac{f(3 x)}{f(x)}=1\) then \(\lim _{x \rightarrow \infty} \frac{f(2 x)}{f(x)}=\) [2010] (a) \(\frac{2}{3}\) (b) \(\frac{3}{2}\) (c) 3 (d) 1

Step-by-Step Solution

Answer

The limit is 1 (Option d).

Step 1: Understand the Given Condition

We are given that the limit \( \lim _{x \rightarrow \infty} \frac{f(3 x)}{f(x)}=1 \). This implies that as \( x \) approaches infinity, the ratio of \( f(3x) \) to \( f(x) \) approaches 1, suggesting that the rate of growth of \( f(3x) \) is similar to \( f(x) \).

Step 2: Assume a Possible Behavior of f(x)

Given the condition, assume a function (for the sake of exploration) such as \( f(x) = x^a \) where \( a > 0 \). This is a common strategy to test the given functional behavior due to its simplicity.

Step 3: Apply Assumed Function to the Given Condition

Calculate \( \frac{f(3x)}{f(x)} \) with \( f(x) = x^a \). We have: \[ \frac{f(3x)}{f(x)} = \frac{(3x)^a}{x^a} = 3^a. \] Since \( \lim _{x \rightarrow \infty} \frac{f(3x)}{f(x)} = 1 \), it follows that \( 3^a = 1 \). Thus, \( a = 0 \). But this would imply a constant function, which is not increasing and positive. So instead, consider that the form might generally behave similarly without specific exponent conclusions.

Step 4: Explore Implications for f(2x)

Using a similar reasoning as with \( f(3x) \), we compute \( \frac{f(2x)}{f(x)} \). Assuming the growth is similar around constant multiples, it follows that \( \lim_{x \rightarrow \infty} \frac{f(2x)}{f(x)} = c \), where \( c \) should be similar to the pattern given, suggesting \( c=1 \).

Step 5: Conclude Using Problem's Asymptotic Condition

The condition suggests that the growth from scaling by any constant factor is nullified asymptotically to 1. Thus, \( \lim_{x \rightarrow \infty} \frac{f(2x)}{f(x)} = 1 \) should hold true because the limit condition holds for tripling as a similar principle applies for doubling without the specific function form revealing otherwise.

Key Concepts

Asymptotic behavior

Asymptotic behavior refers to the way a function behaves as it approaches a particular point, often at infinity. In the context of the given exercise, examining the asymptotic behavior involves understanding what happens to the ratio \( \frac{f(3x)}{f(x)} \) as \( x \) approaches infinity. This specific condition implies that as \( x \) grows very large, the comparative difference between \( f(3x) \) and \( f(x) \) diminishes to nearly zero, leading the limit to approach 1.

• An asymptotic expression helps identify if two functions grow at the same rate for large values of \( x \).
• By analyzing the behavior when \( x \to \infty \), we effectively discard minor fluctuations in favor of understanding the overarching trend.

Identifying this asymptotic condition aids in interpreting the function's behavior. It indicates that scaling \( x \) by a constant factor such as 3 doesn't affect the relative growth trajectories of \( f(3x) \) and \( f(x) \) as they both head towards infinity. In essence, as \( x \to \infty \), \( f(x) \) scales in such a way that this condition of \( \frac{f(3x)}{f(x)} = 1 \) is satisfied.

Function growth rate

The concept of function growth rate concerns how rapidly a function increases as its input becomes large. The growth rate is a central idea in comparing how functions behave, particularly when multiplied by a constant factor. In this problem, the expression \( \frac{f(3x)}{f(x)} \to 1 \) suggests that \( f(3x) \) and \( f(x) \) increase at nearly the same rate.

• It implies that when \( x \) triples, \( f(x) \) effectively compensates for the change to maintain the proportion.
• A similar growth pattern is expected when analyzing \( \frac{f(2x)}{f(x)} \), indicating consistent proportional increases regardless of the factor.

Finding \( \frac{f(2x)}{f(x)} \to 1 \) suggests symmetry in the function's growth. This means changing \( x \) linearly by any constant factor leads to a consistent relative increase in \( f(x) \). The growth is demonstrated directly through limits, illustrating how \( f(2x) \) should similarly align with \( f(x) \) in terms of their asymptotic tendencies, maintaining a function growth rate that balances over any constant scaling.

Limit concepts

Limit concepts are fundamental in calculus, capturing how a function behaves near a certain point or as inputs grow indefinitely. Here, the limit \( \frac{f(3x)}{f(x)} \to 1 \) translates to the function \( f(x) \) maintaining a stable growth rate as \( x \to \infty \).

• Limits help navigate how function values evolve, pinning down that even as inputs swell, the growth doesn't skew excessively in either direction relative to scaling.
• This principle is extrapolated to guess similar behavior for \( \frac{f(2x)}{f(x)} \), i.e., as \( x \) doubles, \( f(2x) \) should mirror \( f(x) \) closely enough for the limit to still equal 1.

Limit concepts support the hypothesis that scale shifts in any constant way do not change asymptotic ratios if originally shown consistent through a specific example \((3x)\). This frames central assumption boundaries in function behavior, allowing educators to generalize known results as seen in forms of the similar expression \( \frac{f(2x)}{f(x)} \to 1 \), emphasizing stability across varying constants.