Problem 320 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \lim _{x \rightarrow \infty} \frac{\ln x}{x}\\{\text { Ans. } 0\\} $$

Step-by-Step Solution

Answer

Using L'Hôpital's Rule, we differentiate both the numerator (\(\ln x\)) and the denominator (\(x\)) to get \(u'(x) = \frac{1}{x}\) and \(v'(x) = 1\). Then we evaluate the limit of the ratio of the derivatives: \(\lim_{x \rightarrow \infty} \frac{u'(x)}{v'(x)} = \lim_{x \rightarrow \infty} \frac{\frac{1}{x}}{1}\). The limit of this expression is 0, thus the answer to the original limit problem is: \(\lim_{x \rightarrow \infty} \frac{\ln x}{x} = 0\).

Step 1: Identify L'Hôpital's Rule applicability

First, we need to analyze the function \(\frac{\ln x}{x}\) as \(x\) approaches infinity. Both the numerator (\(\ln x\)) and the denominator (\(x\)) tend to infinity as \(x\) approaches infinity, so the limit is in an indeterminate form. That means we can apply L'Hôpital's Rule to find the limit.

Step 2: Differentiate the numerator and denominator

Next, we need to differentiate both the numerator and the denominator with respect to \(x\). So, we get: \(u(x) = \ln x\) \(v(x) = x\) \(u'(x) = \frac{1}{x}\) \(v'(x) = 1\)

Step 3: Apply L'Hôpital's Rule

Now we can apply L'Hôpital's Rule and find the limit of the ratio of the derivatives: \(\lim_{x \rightarrow \infty} \frac{u'(x)}{v'(x)} = \lim_{x \rightarrow \infty} \frac{\frac{1}{x}}{1}\)

Step 4: Simplify the limit

After applying L'Hôpital's Rule, we should simplify the limit expression. In this case, we just need to find the limit of \(\frac{1}{x}\) as \(x\) approaches infinity: \(\lim_{x \rightarrow \infty} \frac{1}{x} = 0\)

Step 5: Conclude the answer

Since we found the limit of the ratio of derivatives, we can conclude that the answer to the original limit problem is: \(\lim_{x \rightarrow \infty} \frac{\ln x}{x} = 0\)

Key Concepts

L'Hôpital's Rule

L'Hôpital's Rule is a powerful mathematical tool for finding limits of ratios of functions that result in indeterminate forms. It simplifies the problem by allowing us to differentiate the numerator and the denominator separately. This rule comes in handy, especially when direct substitution in a limit problem gives

• \(\frac{\infty}{\infty}\)
• \(\frac{0}{0}\)

L'Hôpital's Rule tells us to take the derivative of the top function and take the derivative of the bottom function. Then, compute the limit of these new derivatives instead.
Be mindful that L'Hôpital's Rule can only be used when the limits of both the numerator and denominator tend to an indeterminate form. Furthermore, ensure that the new limit of the derivatives exists or can be calculated.

Indeterminate Forms

Indeterminate forms occur in mathematics when the direct substitution of a value into an expression yields an unclear result like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). These forms indicate that we need a different strategy or more analysis to find the precise limit.
Mathematicians encounter these forms frequently, especially when studying functions at their boundaries, like in limits approaching infinity. It's a signal that we can't draw a conclusion about the limit just from the basic arithmetic of fractions.
Several approaches exist to resolve indeterminate forms:

• Using algebraic simplification to clear terms
• Applying L'Hôpital's Rule when applicable
• Rewriting the expression using equivalent forms

Understanding indeterminate forms helps in identifying when to switch gears in calculation strategies, ensuring proper usage of deeper mathematical tools.

Differentiation

Differentiation is a core concept in calculus, focusing on finding the rate at which a function changes. Imagine tracing the slope of a tangent to the curve of a function; that slope is what differentiation aims to calculate.
For any function \(f(x)\), the derivative is denoted by \(f'(x)\) or \(\frac{df}{dx}\). It describes the instantaneous rate of change of \(f(x)\) with respect to \(x\).
When applying L'Hôpital's Rule, we differentiate the functions involved in a fraction. This means:

• The numerator becomes one function to differentiate.
• The denominator becomes another for differentiation.

Differentiation helps us simplify complex limits, turning them into a more manageable form. It's crucial in transforming functions whose limits result in indeterminate forms, paving the way to clearer analysis.

Natural Logarithm

The natural logarithm, denoted by \(\ln x\), is a logarithm to the base \(e\), where \(e\) is approximately 2.718. It's particularly useful in calculus because it transforms multiplicative processes into additive ones, making complex calculations simpler.
In limit problems, functions involving \(\ln x\) often arise, as seen in our example \(\frac{\ln x}{x}\). The natural logarithm has good properties:

• It grows slowly compared to polynomial functions as \(x\) approaches infinity.
• Its derivative is \(\frac{1}{x}\), which frequently simplifies expressions in calculus.

Understanding how \(\ln x\) behaves at infinity helps to contrast it with other functions, like \(x\) in our example, leading to better insights into the limit's behavior. This contrast often reveals why \(\ln x\) in the numerator leads the whole fraction to zero as \(x\) grows large, due to its slower growth rate.