Problem 135 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(f: R \rightarrow R\) be a positive increasing function such that \(\lim _{x \rightarrow \infty} \frac{f(3 x)}{f(x)}=1 .\) Then, \(\lim _{x \rightarrow \infty} \frac{f(2 x)}{f(x)}=\) (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 Limit Condition

We are given that \( \lim_{x \to \infty} \frac{f(3x)}{f(x)} = 1 \). This means that for very large values of \( x \), \( f(3x) \) and \( f(x) \) grow at the same rate.

Step 2: Hypothesize Form of Solution

Since \( f(3x) \sim f(x) \), it implies that \( f(x) \) may be in the form of an exponential function \( f(x) = x^a \), where the power \( a \) will help satisfy the limit. However, for our purposes and due to the nature of growth, we can also use the information to guess that \( \frac{f(kx)}{f(x)} \to 1 \) implies the function behaves polynomially.

Step 3: Find Relationship using the Limit

Given \( \frac{f(3x)}{f(x)} \to 1 \), the behavior of \( f \) suggests that multiplying the input by a constant affects the output proportionally, so:\[ f(3x) \sim c f(x) \] for some constant \( c \) such that \( c = 1 \). Therefore, exploring the behavior as \( x \to \infty \) when replacing 3 with 2 should follow the same notion.

Step 4: Compute \( \frac{f(2x)}{f(x)} \)

Now, considering the similar argument for \( \lim_{x \to \infty} \frac{f(2x)}{f(x)} \):\[ \lim_{x \to \infty} \frac{f(2x)}{f(x)} \] Since \( f(3x) \sim f(x) \):\[ f(3x) = c f(x) \] with \( c = 1 \), we conjecture the same behavior for \( f(2x) \) and find it behaves similarly and thus \( \frac{f(2x)}{f(x)} \rightarrow 1 \). However, we need to conclude this logically with an assumption of the direct implication of rates as for all rates; polynomial growth implicitly follows (e.g., if \( x \to ax \), then same behavior is expected \( c = 1 \) due to limits).

Step 5: Confirm the Result with Logic Reasoning

Given the condition \( \frac{f(3x) \sim f(x)} \), \( f(3x) \) representing same growth as \( f(x) \) strongly suggests that any \( kx \) (here \( 2x \)) similarly must logically lead to similar growth parameters, due to \( x \to kx \) would not diverge as value (rate axis), ensuring \( \frac{f(2x)}{f(x)} \to 1 \) as well.

Key Concepts

Positive Increasing Function

A positive increasing function is a function where values consistently rise as the input increases. In mathematical terms, this looks like:

• If you have two numbers, say, \(x_1\) and \(x_2\), and \(x_1 < x_2\), then for an increasing function \(f\), \(f(x_1) < f(x_2)\).
• This function never dips below the horizontal axis, meaning \(f(x) > 0\) for all \(x\). This is what being positive means.

Positive increasing functions appear in many practical scenarios, like functions representing growing populations or ever-increasing bank interest.
In this problem, the function \(f\) is not only positive and increasing but also has a special property: the limit as \(x\) approaches infinity of the quotient \(\frac{f(3x)}{f(x)} = 1\). This tells us that although \(f(3x)\) becomes much larger than \(f(x)\) as \(x\) increases, their relative rates of growth are effectively the same as \(x\) grows larger and larger.

Limit of a Function

Understanding the limit of a function is crucial in calculus. The limit describes the behavior of a function as its input approaches a certain value. Here, we're looking at how \(f(x)\) behaves as \(x\) goes to infinity, which means an exceedingly large value.
When we are given that \[\lim_{x \to \infty} \frac{f(3x)}{f(x)} = 1\]it signifies that for very large \(x\), \(f(3x)\) and \(f(x)\) grow at such a rate that their ratio tends toward 1. This suggests that the function \(f(x)\) might grow in a manner where stretching the input by a factor of 3 does not change the overall growth behavior relative to \(f(x)\).
For the given exercise, we use the suggested behavior of limits to predict that when the same function is applied to \(2x\), instead of \(3x\), it should show similar growth characteristics. By understanding limits, one can infer that since \(\frac{f(3x)}{f(x)}\) approaches 1, then \(\frac{f(2x)}{f(x)}\) should also tend to 1, demonstrating uniform growth behavior.

Polynomial Growth Behavior

Polynomial growth refers to situations where a function increases at a rate proportional to some power of the variable. If a function \( f(x) = x^a \), it exhibits polynomial growth, and the power \( a \) determines the growth rate.

• A higher power \( a \) indicates faster growth.
• Functions like \( x^2 \) or \( x^3 \) grow faster than \( x \) itself as \( x \) becomes very large.

In this context, the exercise involves understanding that the function \( f \), which we hypothesize to be polynomial due to its behavior under limits, doesn't expand significantly faster than any linear multiplier of \( x \), due to the limit provided.
So, when analyzing the function with values like \(3x\) and \(2x\), the similar results from the limits suggest polynomial behavior. The limit \[\frac{f(3x)}{f(x)} \to 1\]implies \(f\) shares similar growth behavior over increasing scales. This suggests that "scaling up" through the factor of a constant preserves the growth relation (like making both sides of a scale larger). Thus, transitioning from \(3x\) to \(2x\), still sees this polynomial-like effect, leading us to conclude the same outcome \(\frac{f(2x)}{f(x)} \to 1\). The limit highlights constant growth properties, typical of polynomial behavior.