Problem 356 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \lim _{x \rightarrow 0} \frac{a^{x}-b^{x}}{c^{x}-d^{x}}\left\\{\text { Ans. } \frac{\ln \frac{a}{b}}{\ln \frac{c}{d}}\right\\} $$

Step-by-Step Solution

Answer

The short answer is: \(\frac{\ln \frac{a}{b}}{\ln \frac{c}{d}}\).

Step 1: Identify numerator and denominator functions

The given function is: \(\lim _{x \rightarrow 0} \frac{a^{x}-b^{x}}{c^{x}-d^{x}}\). Let, \(f(x) = a^{x}-b^{x}\) and \(g(x) = c^{x}-d^{x}\). Now, we need to find the derivatives of f(x) and g(x).

Step 2: Differentiate f(x) and g(x) with respect to x

Differentiate f(x) with respect to x: \(f'(x) = \frac{d}{dx} (a^{x} - b^{x}) = a^{x} \ln(a) - b^{x} \ln(b)\). Differentiate g(x) with respect to x: \(g'(x) = \frac{d}{dx} (c^{x} - d^{x}) = c^{x} \ln(c) - d^{x} \ln(d)\). Now we will apply L'Hopital's rule.

Step 3: Apply L'Hopital's Rule to find the limit

Applying L'Hopital's rule for the given function: \(\lim_{x \rightarrow 0} \frac{a^{x}-b^{x}}{c^{x}-d^{x}} = \lim_{x \rightarrow 0} \frac{a^{x} \ln(a) - b^{x} \ln(b)}{c^{x} \ln(c) - d^{x} \ln(d)}\)

Step 4: Evaluate the limit as x approaches 0

Now, let's evaluate the limit as x approaches 0: \(\lim_{x \rightarrow 0} \frac{a^{x} \ln(a) - b^{x} \ln(b)}{c^{x} \ln(c) - d^{x} \ln(d)} = \frac{a^{0} \ln(a) - b^{0} \ln(b)}{c^{0} \ln(c) - d^{0} \ln(d)} = \frac{\ln(a) - \ln(b)}{\ln(c) - \ln(d)}\) To simplify the expression further, we use the property of logarithms that states \(\ln(\frac{a}{b}) = \ln(a) - \ln(b)\). Therefore, the final answer is: \(\boxed{\frac{\ln \frac{a}{b}}{\ln \frac{c}{d}}}\).

Key Concepts

Limit Calculus

Limit calculus is the study of the behavior of a function as the input approaches some value. Understanding limits is crucial to solving many problems in calculus, particularly when evaluating indeterminate forms like \(\frac{0}{0}\). In this example, we try to find:

• \(\lim _{x \rightarrow 0} \frac{a^{x}-b^{x}}{c^{x}-d^{x}}\)

The limit tells us how the fraction behaves as \(x\) gets arbitrarily close to 0. Often, directly substituting \(x = 0\) leads to an indeterminate form, such as \(\frac{0}{0}\), where calculation requires special techniques like L'Hopital's Rule. This method involves differentiating the numerator and the denominator of the fraction and then finding the limit of their quotient. Limits provide a way to understand and define derivative and integral behaviors in functions.

Differentiation

Differentiation is a fundamental concept in calculus that deals with finding the rate at which a function is changing at any given point. Differentiating a function involves calculating its derivative. In this exercise, we have two functions:

• \(f(x) = a^{x}-b^{x}\)
• \(g(x) = c^{x}-d^{x}\)

To apply L'Hopital's Rule, we differentiate these functions with respect to \(x\). The process involves using the chain rule:

• The derivative of \(a^{x}\) is \(a^{x} \ln(a)\)
• The derivative of \(b^{x}\) is \(b^{x} \ln(b)\)

This results in:

• \(f'(x) = a^{x} \ln(a) - b^{x} \ln(b)\)
• \(g'(x) = c^{x} \ln(c) - d^{x} \ln(d)\)

Differentiation provides the tools for re-evaluating the limit to find a clearer expression that avoids indeterminate conditions.

Logarithmic Functions

Logarithmic functions are important in calculus for simplifying expressions, especially using their properties in solving limits and derivatives. In the original problem, the logarithmic properties make final simplification possible:

• \(\ln(\frac{a}{b}) = \ln(a) - \ln(b)\)
• \(\ln(\frac{c}{d}) = \ln(c) - \ln(d)\)

These properties were essential in the transformation of the function’s form in the final answer:

• \(\frac{\ln(a) - \ln(b)}{\ln(c) - \ln(d)} = \frac{\ln \frac{a}{b}}{\ln \frac{c}{d}}\)

Understanding the natural logarithm, denoted \(\ln(x)\), is pivotal, as it relates exponential growth to addition, greatly simplifying the process of differentiation and integration concerning exponential terms in calculus.