Problem 429 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \text { If } \left.f(x)>\ln x \forall x>0, \text { find } \lim _{x \rightarrow \infty} f(x) \text { . Ans. } \infty\right\\} $$

Step-by-Step Solution

Answer

Since \(f(x)>\ln(x)\) for all \(x>0\), and the limit of the natural logarithm as \(x\) tends to infinity is \(\infty\), then the limit of \(f(x)\) as \(x\) approaches infinity must also be \(\infty\). Therefore, \(\lim_{x \rightarrow \infty} f(x) = \infty\).

Step 1: Analyze given information

The statement \(f(x) > \ln x\) for all \(x > 0\) implies that the function f(x) is continuously growing at a faster rate than the natural logarithm, which also grows towards infinity, but at a slower pace.

Step 2: Write inequality for limits

As x tends to infinity, we can write the inequality for limits: \[\lim_{x \rightarrow \infty} f(x) > \lim_{x \rightarrow \infty} \ln(x)\]

Step 3: Evaluate limit of ln(x)

Recall the property of the natural logarithm, which states that as x tends to infinity, the limit of the natural logarithm also tends to infinity: \[\lim_{x \rightarrow \infty} \ln(x) = \infty\]

Step 4: Conclude the limit of f(x)

Since the limit of the natural logarithm of x is infinity, and the function f(x) is always greater than the natural logarithm for all \(x>0\), the limit of f(x) will also tend to infinity as \(x\) approaches infinity. Therefore, we conclude that: \[\lim_{x \rightarrow \infty} f(x) = \infty\]

Key Concepts

Inequality

Understanding inequalities is crucial in analyzing the behavior of functions. An inequality, in a mathematical sense, shows a relationship between two expressions or values. When we see an expression like \( f(x) > \ln x \), it tells us that the function \( f(x) \) always takes values greater than the value of the natural logarithm of \( x \) for every \( x > 0 \).

This concept becomes especially important in calculus when dealing with limits. It can hint at how a function behaves compared to other functions as the input grows. In our original problem, the inequality points out that \( f(x) \) increases faster than \( \ln x \), implying its value will dominate the logarithm function as \( x \) approaches infinity.

To grasp this better, consider that solving inequalities often involves comparing functions and analyzing their rates of growth or decay:

• Functions that grow faster will eventually surpass slower-growing ones.
• Understanding limits involves identifying which part of the inequality becomes dominant with larger values of \( x \).

Grasping these ideas helps us predict the behavior of \( f(x) \) relative to \( \ln x \) as \( x \rightarrow \infty \).

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is the logarithm to the base \( e \), where \( e \) is an irrational and transcendental number approximately equal to 2.71828. The function \( \ln(x) \) is significant in mathematics due to its unique properties and application in different contexts, especially in calculus and analysis.

One of the notable properties of \( \ln(x) \) is its slow growth compared to polynomial and exponential functions. As \( x \) increases, \( \ln(x) \) also increases but at a decelerating rate. This is important when looking at limits as \( x \rightarrow \infty \).

Here's a brief overview of its key features:

• The domain of \( \ln(x) \) is all positive real numbers \( (x>0) \).
• It passes through the point \( (1, 0) \) as \( \ln(1) = 0 \).
• As \( x \rightarrow \infty \), the limit \( \lim_{x \rightarrow \infty} \ln(x) = \infty \).

By understanding these properties, one can better predict how \( \ln(x) \) will behave relative to other functions like \( f(x) \) in our given problem, where \( f(x) > \ln(x) \).

Infinity

Infinity is a concept rather than a number, representing boundlessness or an unending value. In calculus, it is used to describe the behavior of functions or sequences that grow without bounds.

When discussing limits involving infinity, we aim to understand how a function behaves as its input approaches an infinitely large value, such as \( x \rightarrow \infty \). In our context, particularly with \( f(x) > \ln x \) for all \( x > 0 \), we are concerned with predicting what happens to \( f(x) \) as \( x \) grows larger and larger.

The key points to remember about infinity in calculus are:

• It is not a real number and can't be used in ordinary arithmetic operations.
• Not all functions that grow will reach infinity; the rate of growth and the comparison with other functions matter.
• When a function's limit as \( x \rightarrow \infty \) is stated to equal infinity, it means the function grows indefinitely.

Understanding infinity helps interpret the behavior of \( f(x) \) in the problem. Since \( \ln(x) \) itself tends to infinity and \( f(x) \) grows faster than \( \ln(x) \), it leads to the conclusion that \( \lim_{x \rightarrow \infty} f(x) = \infty \). This shows the unbounded nature of \( f(x) \) as \( x \rightarrow \infty \), faster than the natural logarithm ever could.