Problem 347 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \lim _{x \rightarrow 0}(\ln \cot x)^{\tan x}\\{\text { Ans. } 1\\} $$

Step-by-Step Solution

Answer

The limit of the given function as x approaches 0 is 0

Step 1: Identify the indeterminate form

First identify the form of the given function as x approaches 0, which is \((\ln \cot x)^{\tan x}\). We can see it is of the form \(0^0\) as \(\ln \cot 0\) tends to \(-\infty\) and \(\tan 0\) is 0, which makes the expression indeterminate.

Step 2: Simplify the expression

In order to solve it, we can simplify the logarithmetic part of the function by using the relationship \(\ln a^b = b \ln a\), then it becomes \(\tan x \cdot \ln(\cot x)\). Now, we can confirm that as x approaches 0, the expression is of the \(0 \cdot -\infty\) form, which can be simplified into a \(\frac{0}{0}\) or \(\frac{-\infty}{\infty}\) form by putting \(\tan x = \frac{1}{\cot x}\), thus the expression can be written as \(\frac{\ln(\cot x)}{\cot x}\)

Step 3: Apply l'Hopital's Rule

Now that we've transformed our indeterminate form into a quotient, we can apply l'Hopital's rule, which states that for any limit of the form \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), we can take the derivative of both the numerator and the denominator individually until we get a determinable limit. Derivatives of the logarithm function and the cotangent function are \(\frac{1}{x}\) and \(-\csc^2 x\), respectively. Hence, we have \(\lim_{x \to 0} \frac{1/\cot x}{-\csc^2 x}\), which simplifies to \(\lim_{x \to 0} -\tan x = 0\)

Step 4: Evaluate the limit

After simplification, our function becomes a determinate form. So now, we can apply the limit and get the final result. Thus, the value of the limit as x tends to 0 is 0

Key Concepts

Indeterminate Forms

Indeterminate forms occur in calculus when evaluating a limit doesn't initially yield a clear outcome. They are called 'indeterminate' because it's not immediately obvious what value they will resolve to, which necessitates the use of further analysis or mathematical techniques.

• The common types of indeterminate forms include: \(\frac{0}{0}\), \(\frac{\infty}{\infty}\), \(0\cdot\infty\), \(\infty - \infty\), \(1^\infty\), \(0^0\), and \(\infty^0\).
• In our exercise, the expression \((\ln \cot x)^{\tan x}\) as \(x\) approaches 0 appeared initially to be of form \(0^0\), transforming into a \(0 \cdot -\infty\) form during simplification. These forms complicate direct evaluation of limits.

Indeterminate forms leave us with ambiguity regarding the limit's final result, prompting additional methods to find the correct value. Understanding them is crucial as they often appear in calculus problems.

L'Hopital's Rule

L'Hopital's Rule is a powerful tool in calculus used to resolve limits involving indeterminate forms, particularly the \(\frac{0}{0}\) and \(\frac{\infty}{\infty}\) types. The rule allows us to simplify these forms by differentiating the numerator and the denominator separately.

• To apply L'Hopital's Rule, transform the indeterminate form into a fraction where you get one of the applicable forms \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\).
• Differentiate both the numerator and denominator independently, and then compute the limit of this new function.
• In our specific exercise, transforming the expression to \(\frac{\ln(\cot x)}{\cot x}\) allowed us to use L'Hopital's Rule as both components tended toward forms suitable for its application.

The key advantage of L'Hopital's Rule is that it often circumvents tedious algebraic manipulations, directly bringing the problem to a simpler form. **Note:** Sometimes multiple applications of the rule, or even alternative methods, are necessary if the new function still presents an indeterminate form.

Logarithmic Functions

Logarithmic functions, most simply understood as the inverse of exponential functions, play a significant role in calculus. Here, they help in dealing with exponents and complexities involving growth, decay, and trends.

• The logarithm base \(b\), usually denoted as \(\log_b(x)\), answers the question: to what power must \(b\) be raised, to yield \(x\)? Most commonly used bases are 10 and \(e\) (natural logarithms), with \(\ln(x)\) denoting the natural logarithm.
• In calculus, logarithmic differentiation often simplifies the process of finding limits, especially when dealing with exponentials involved in a given function.
• In our exercise, simplifying the function using logarithmic identities, like \(\ln(a^b) = b \ln(a)\), transformed a complex power into a solvable product.

Understanding logarithmic functions is crucial as they frequently appear across diverse mathematical scenarios, including solving equations, transforming data, and analyzing limits. They simplify complex expressions, making intricate derivations more manageable.