Problem 335 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \lim _{x \rightarrow 0} x^{\frac{1}{\ln \left(e^{x}-1\right)}}\\{\text { Ans. } e\\} $$

Step-by-Step Solution

Answer

To find the limit \(\lim _{x \rightarrow 0} x^{\frac{1}{\ln \left(e^{x}-1\right)}}\), we first rewrite the expression as \(\lim _{x \rightarrow 0} e^{\frac{\ln(x)}{\ln \left(e^{x}-1\right)}}\). Then, we define the limit in terms of indeterminate form 0/0 as \(f(x) = \lim _{x \rightarrow 0}\frac{\ln(x)}{\ln \left(e^{x}-1\right)}\). Next, we apply L'Hopital's rule twice, first on \(f(x)\) and then on the simplified expression \(\lim _{x \rightarrow 0} \frac{e^x - 1}{x}\), obtaining \(f(x) = 1\). Finally, we substitute the value of f(x) back into the original expression, resulting in the limit being equal to \(e^1 = e\).

Step 1: Rewrite the given expression using logarithms

First, we should rewrite the given expression into a more suitable form for applying L'Hopital's rule. We can use the following property of logarithms: \(a^b = e^{\ln(a^b)}\). So, we have \(\lim _{x \rightarrow 0} x^{\frac{1}{\ln \left(e^{x}-1\right)}} = \lim _{x \rightarrow 0} e^{\frac{\ln(x)}{\ln \left(e^{x}-1\right)}}\).

Step 2: Define the limit in terms of indeterminate form 0/0

In order to apply L'Hopital's rule, we need the limit to be an indeterminate form of 0/0. We can see that both the numerator and denominator of the exponent go to 0 as x approaches 0. So, let's rewrite the expression as: \(f(x) = \lim _{x \rightarrow 0}\frac{\ln(x)}{\ln \left(e^{x}-1\right)}\). Now, we can apply L'Hopital's rule to f(x).

Step 3: Apply L'Hopital's rule

Applying L'Hopital's rule, we differentiate both the numerator and denominator with respect to x: \(f(x) = \lim _{x \rightarrow 0} \frac{\frac{d}{dx} \ln(x)}{\frac{d}{dx} \ln \left(e^{x}-1\right)}\) Using the chain rule we have: \(f(x) = \lim _{x \rightarrow 0} \frac{\frac{1}{x}}{\frac{e^x}{e^x - 1}}\)

Step 4: Evaluate the limit

Now, as x approaches 0, we find the limit: \(f(x) = \lim _{x \rightarrow 0} \frac{\frac{1}{x}}{\frac{e^x}{e^x - 1}} = \lim _{x \rightarrow 0} \frac{e^x - 1}{x}\) We can apply L'Hopital's rule one more time: \(f(x) = \lim _{x \rightarrow 0} \frac{\frac{d}{dx}(e^x - 1)}{\frac{d}{dx}x} = \lim _{x \rightarrow 0} \frac{e^x}{1}\) So, as x approaches 0: \(f(x) = 1\)

Step 5: Go back to the original expression

Now that we have found the limit of f(x), we can use it to find the limit of the original expression: \(\lim _{x \rightarrow 0} e^{\frac{\ln(x)}{\ln \left(e^{x}-1\right)}} = e^{f(x)}\\ = e^1 = e\) Therefore, the limit as x approaches 0 for the given expression is e.

Key Concepts

Indeterminate Forms

In calculus, indeterminate forms arise when evaluating limits of functions, and the limit expression initially provides an uncertain outcome. A common example is the expression \( \frac{0}{0} \), which occurs when both the numerator and denominator of a fraction approach zero as the variable approaches a certain value. Other indeterminate forms include \( \frac{\infty}{\infty} \), \( 0 \times \infty \), \( \infty - \infty \), \( 0^0 \), \( \infty^0 \), and \( 1^\infty \). These forms are called indeterminate because they do not lead to a unique limit; various functions that exhibit the same form when approaching a limit can have different limit values.

To handle indeterminate forms, mathematicians have developed several techniques, including factorization, conjugation, and the widely-used L'Hopital's rule. It is important to understand that indeterminate forms are merely a signal that further analysis is needed to find the limit, rather than a final answer in themselves.

L'Hopital's Rule

When faced with an indeterminate form like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), L'Hopital's rule offers an effective method for finding limits. To apply this rule, we take the derivatives of the numerator and the denominator separately and then evaluate the limit of the resulting expression. If the limit remains an indeterminate form, L'Hopital's rule can be reapplied.

It is crucial, before applying L'Hopital's rule, to confirm that the original limit does indeed result in an indeterminate form. L'Hopital's rule can streamline the solution process significantly when used appropriately. However, its application is not universal and must be justified; it only applies to the specific indeterminate forms for which it is designed. Always check the conditions for applying this rule carefully, including the requirement that the derivatives of the numerator and denominator must exist around the point of interest.

Exponential Functions

Using exponential functions is a common procedure in solving various mathematical problems, particularly in calculus. An exponential function has the form \( f(x) = a^x \), where \( a \) is a constant, and \( a > 0 \). Exponential functions exhibit unique properties that make them invaluable when modeling growth or decay processes in fields such as finance, physics, and biology.

One notable characteristic of exponential functions is their derivative. The function \( e^x \), where \( e \) is Euler's number (approximately 2.71828), is particularly significant because it is the only function that is equal to its own derivative. This property is vital in solving many calculus problems. The limit of exponential functions as \( x \) approaches zero is also noteworthy, as \( \lim_{x \to 0} e^x = 1 \), a fact that is often useful when evaluating limits involving exponentials. Understanding the behavior and manipulation of these functions is essential for solving a wide range of mathematical and real-world problems.