Problem 206 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

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

Step-by-Step Solution

Answer

The short answer for the given limit problem is: \(\lim_{x\rightarrow 0}(1+ax)^{\frac{b}{x}} = e^{ab}\).

Step 1: Identify the given problem

We are given the limit problem: \[ \lim_{x\rightarrow 0}(1+ax)^{\frac{b}{x}} \]

Step 2: Recognizing the indeterminate form

As x approaches 0, we have the form \(1^{\infty}\), which is an indeterminate form. To handle this situation, we will take the natural logarithm (ln) of the function and then use L'Hopital's Rule.

Step 3: Take the natural logarithm of the function

Take the logarithm of the function, and then we can use properties of logs to rewrite it as follows: \[ \ln\left( \lim_{x\rightarrow 0}(1+ax)^{\frac{b}{x}} \right)= \lim_{x\rightarrow 0}\left(\frac{b}{x}\ln(1+ax)\right) \]

Step 4: Use L'Hopital's Rule

Use L'Hopital's Rule for the transformed limit. Differentiate both the numerator and the denominator with respect to x: \[ \lim_{x\rightarrow 0}\frac{\frac{d}{dx}(b\ln(1+ax))}{\frac{d}{dx}(x)} \] Using the chain rule, we get the derivative of the numerator as: \(\frac{d}{dx}(b\ln(1+ax)) = b\cdot\frac{a}{1+ax}\) Now apply L'Hopital's Rule: \[ \lim_{x\rightarrow 0}\frac{b\cdot\frac{a}{1+ax}}{1} = \lim_{x\rightarrow 0}(b\cdot\frac{a}{1+ax}) \]

Step 5: Evaluate the limit

As x approaches 0, we have: \[ \lim_{x\rightarrow 0}(b\cdot\frac{a}{1+ax}) = b\cdot\frac{a}{1+0} = ab \] We took the natural logarithm in Step 3, so now we need to exponentiate the result: \[ e^{\lim_{x\rightarrow 0}\ln\left((1+ax)^{\frac{b}{x}}\right)} = e^{ab} \]

Step 6: Write the final answer

The final answer for the given limit problem is: \[ \lim_{x\rightarrow 0}(1+ax)^{\frac{b}{x}} = e^{ab} \]

Key Concepts

Indeterminate Forms

In calculus, not all expressions lead to straightforward limits. An indeterminate form is a key concept where the limit is not immediately apparent. As the name suggests, these forms do not directly determine an answer and can lead to a variety of outcomes depending on the specific mathematical situation.

There are a few common types of indeterminate forms, like:

• \(0/0\), which often happens in rational functions.
• \(\infty / \infty\), which can occur when both the numerator and denominator approach infinity.
• \(1^{\infty}\), the scenario in our problem, representing an unclear result when a number slightly greater than 1 is raised to an infinitely large power.

Understanding when you're facing an indeterminate form is crucial. It's the first step in deciding which advanced mathematical techniques, such as L'Hopital's Rule, are necessary to resolve the limit.

L'Hopital's Rule

L'Hopital's Rule is a powerful tool in calculus used to tackle indeterminate forms like \(\frac{0}{0}\) and \(\frac{\infty}{\infty}\). This rule is particularly helpful because it allows us to simplify complicated limits by differentiating the numerator and the denominator until we get a determinate form.

To use L'Hopital's Rule properly, follow these key steps:

• Confirm that the limit gives an indeterminate form.
• Differentiate the numerator and denominator separately.
• Re-evaluate the limit with the new expression.

It's important to note that L'Hopital's Rule can only be applied under specific conditions. The functions involved must be differentiable, and we use the rule when direct substitution doesn't work. In our exercise, L'Hopital's Rule is applied after recognizing \(1^{\infty}\), a unique indeterminate form, by manipulating it into a fraction \(\frac{0}{0}\) form using logarithms.

Exponential Functions

Exponential functions are among the most important types of functions in mathematics. They describe processes where growth or decay occurs at a constant relative rate. The general form is \(f(x) = a^x\), where \(a\) is the base and \(x\) is the exponent.

In our problem, exponential functions appear in a slightly more complex way as \((1 + ax)^{\frac{b}{x}}\). As \(x\) approaches 0, this expression outlines how small, incremental changes can accumulate through exponentiation.

Why are they crucial?

• Exponential growth and decay are found in natural phenomena such as population growth, radioactive decay, and interest compounding.
• The number \(e\) (approximately 2.71828) is a fundamental base of natural exponential functions. It arises naturally in limit problems and continuous growth models.

In the context of limit calculus, pitching expressions like \((1 + ax)^{\frac{b}{x}}\) against \lim_{x \rightarrow 0} often makes use of the properties of \(e\). In our problem, converting the expression into \(e^{ab}\) points out how exponential calculation is inherent in solving the limit.