Problem 35 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(\lim _{x \rightarrow 0} \frac{\left(1+a^{3}\right)+8 e^{1 / x}}{1+\left(1-b^{3}\right) e^{1 / x}}=2\), then (A) \(a=1, b=(-3)^{1 / 3}\) (B) \(a=1, b=3^{1 / 3}\) (C) \(a=-1, b=-(3)^{1 / 3}\) (D) None of these

Step-by-Step Solution

Answer

D: None of these.

Step 1: Analyzing the Limit

The limit given is \( \lim _{x \rightarrow 0} \frac{\left(1+a^{3}\right)+8 e^{1 / x}}{1+\left(1-b^{3}\right) e^{1 / x}}=2 \). The expression involves exponential terms of the form \( e^{1/x} \), which tends to infinity as \( x \to 0^- \) and is undefined as \( x \to 0^+ \). Therefore, to satisfy the limit condition, the exponential terms must cancel, indicating that both the numerator and the denominator must equal zero as \( x \to 0^- \), making their highest powers dominate or cancel.

Step 2: Equating Exponential Parts for Global Balance

At \( x \to 0^- \), we need the highest power term in the denominator and numerator to cancel for the limit to exist and be finite. As \( e^{1/x} \to \infty \), both \( 8e^{1/x} \) in the numerator and \( (1-b^3)e^{1/x} \) in the denominator must be of the same order, which means \( 1-b^3 = 8 \), giving \( b^3 = -7 \). However, \( b \) should make the rest of the expression meaningfully contribute to a finite term.

Step 3: Solving for Terms not Involving Exponentials

With the cancellation of exponential terms, the remaining coefficients define the limit value. Thus, set the reduced limit \( \frac{1 + a^3}{1} = 2 \). Solving gives \( 1 + a^3 = 2 \), which implies \( a^3 = 1 \), making \( a = 1 \). For the denominator, \( 1-b^3 = 8 \) gives us \( b^3 = -7 \). The values of \( b \) are the real cube roots of -7, none of which directly match provided options for \( b \).

Step 4: Compare Results to Options

From the above steps, we found \( a = 1 \) and \( b^3 = -7 \). None of the values for \( b = (-7)^{1/3} \) align with the solutions given. Therefore, \( a = 1 \) and \( b \) doesn't satisfy any of the choices. This leads us to verify all the constraints; multiplying through, there are no real given options that hold the balance needed.

Key Concepts

Exponential Functions

Exponential functions are incredibly important in mathematics and science because they describe processes that change at a constant relative rate. At the core of an exponential function is the basic form:

• \( f(x) = a \, e^{bx} \)

where \( a \) and \( b \) are constants, and \( e \) is the base of the natural logarithms, approximately equal to 2.71828. These functions show up in many real-world scenarios, such as population growth, radioactive decay, and continuously compounding interest.

In the context of calculus, exponential functions are unique because their rates of change are proportional to the function itself. This property is central to problems involving differential equations and is also a key factor when assessing limits involving exponential terms, as seen in the exercise above.

Exponential growth or decay can significantly affect the outcome of a limit, especially as the variable approaches a particular value. For instance, terms such as \( e^{1/x} \) can grow extremely fast as \( x \to 0^{-} \), making it necessary to ensure that such large terms are carefully balanced in both the numerator and the denominator in a limit problem.

Rational Functions

Rational functions are ratios of polynomials, defined as

• \( f(x) = \frac{P(x)}{Q(x)} \)

where \( P(x) \) and \( Q(x) \) are polynomials. They are essential in calculus because they represent a fundamental type of function that demonstrates both simplicity and complexity in different contexts.

The exercise involves a rational function where understanding the behavior of both the numerator and the denominator is crucial. In calculus, when evaluating limits involving rational functions, we must carefully consider if the polynomials result in a zero-over-zero indeterminate form, which occurs when both the numerator and denominator approach zero.

This indeterminate form requires further analysis, often by simplifying, factoring, or applying limit laws, to find meaningful solutions. For the given exercise, the challenge was to ensure that the high growth rate of the exponential terms in both the numerator and the denominator canceled out to allow the limit to reach its target value.

Limit Definition

Understanding limits is fundamental to calculus, providing a framework for analyzing how functions behave as inputs get very close to a specific point. The formal definition of a limit states:

• We say \( \lim_{x \to a} f(x) = L \) if for every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that \( 0 < |x - a| < \delta \) implies \( |f(x) - L| < \epsilon \).

In simpler terms, as \( x \) gets closer and closer to \( a \), the function \( f(x) \) approaches the value \( L \).

In practice, especially with limits involving complex functions like the one in this exercise, we often contemplate the behavior of the function as \( x \) approaches from either direction. Limits such as \( \lim_{x \rightarrow 0} \frac{\left(1+a^{3}\right)+8 e^{1 / x}}{1+\left(1-b^{3}\right) e^{1 / x}} = 2 \) involve sorting out the dominant terms. By doing this, you can discover what likely dictates the behavior of the total function around the critical point, allowing for better understanding and finding solutions.