Problem 94 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(f(x)=g^{\prime}(x) \frac{e^{1 / x}-e^{-1 / x}}{e^{y_{x}}+e^{-1 / x}}\), where \(g^{\prime}\) is the derivative of \(g\) and is a continuous function, then \(\lim _{x \rightarrow 0} f(x)\) exists if (A) \(g(x)\) is a polynomial (B) \(g(x)=x\) (C) \(g(x)=x^{2}\) (D) \(g(x)=x^{3} h(x)\), where \(h(x)\) is a polynomial

Step-by-Step Solution

Answer

The limit exists if \( g(x) = x^3 h(x) \), option (D).

Step 1: Analyzing the Function

The given function is \( f(x) = g'(x) \cdot \frac{e^{1/x} - e^{-1/x}}{e^{y_x} + e^{-1/x}} \). To find \( \lim_{x \to 0} f(x) \), we need to understand the behavior of each component as \( x \to 0 \). Note that \( g'(x) \) is continuous, and the expression \( \frac{e^{1/x} - e^{-1/x}}{e^{y_x} + e^{-1/x}} \) will require special attention because of the exponential terms.

Step 2: Simplifying the Exponential Expressions

As \( x \to 0 \), the exponential terms \( e^{1/x} \) and \( e^{-1/x} \) become problematic because \( 1/x \to \infty \) and \(-\infty\) respectively. Investigate the limit using substitution like \( t = 1/x \), transforming it into \( f(1/t) = g'(1/t) \cdot \frac{e^{t} - e^{-t}}{e^{y_{1/t}} + e^{-t}} \). The key is ensuring the denominator handles extreme values of \( e^t \) in such a way that \( f(x) \) remains finite.

Step 3: Determine Conditions on g(x)

For the limit to exist, \( f(x) \) should not depend on the large values produced by exponential terms. This means \( g'(x) \) must offset any behavior caused by \( e^{1/x} - e^{-1/x} \). Each option represents a different polynomial form or product. We determine if these forms allow \( f(x) \) to remain bounded as \( x \to 0 \). Specifically, replacements where \( g(x) = x^n \) (for n > 1) can counteract the behavior of the exponentials.

Step 4: Apply Limit Laws

Evaluate the effect of \( g(x) = x, x^2, x^3 h(x) \) on continuity and boundedness.\( g(x) = x \) and \( g(x) = x^2 \) lead to \( g'(x) = 1 \) and \( g'(x) = 2x \), respectively; these aren't sufficient to offset \( e^{1/x} - e^{-1/x} \). However, \( g(x) = x^3 h(x) \) leads to \( g'(x) = 3x^2 h(x) + x^3 h'(x) \), which, being cubic, has the potential to counterbalance the expressions as \( x \to 0 \).

Step 5: Final Conclusion

Based on our analysis, the limit exists if \( g(x) = x^3 h(x) \) as \( x^2 \) in the derivative gives sufficient terms to counter the exponential behavior in the numerator. This satisfies the balance needed to ensure \( f(x) \) remains bounded and continuous around \( x \to 0 \). Therefore, option (D) is correct.

Key Concepts

Derivative

The concept of a derivative is central to calculus, serving as a foundation for understanding rates of change. In simple terms, the derivative of a function, denoted as \( g'(x) \), represents the slope of the function at a given point. This can be thought of as the instantaneous rate of change as opposed to the average rate of change over an interval.

When solving problems involving limits, especially when exponential functions are involved, the behavior of the derivative can play a key role. If a function is differentiable, it means there's a tangent at every point within its domain, which can greatly aid in evaluating limits.

• The derivative provides insight into how fast or slow a function is changing.
• It remains foundational in determining the behavior of functions near specific points, such as \( x=0 \) in this exercise.
• Polynomials and their derivatives often simplify intricate functions by transforming exponential challenges into more manageable polynomial terms.

Exponential Functions

Exponential functions have the form \( e^x \), where the base \( e \) is a mathematical constant approximately equal to 2.71828. This type of function grows exponentially, which means growth increases rapidly as the variable increases.

In the context of limits in calculus, exponential functions become significant when approaching infinity or zero. As \( x \to 0 \) for the function \( e^{1/x} \), it tends towards an exponential explosion, making it difficult for other parts of the function to balance out these extreme values.

• Exponential terms like \( e^{1/x} \) can dominate the behavior of a function around \( x=0 \).
• Understanding how exponential terms interact, such as \( e^{1/x} - e^{-1/x} \), can provide insights into whether a limit exists.
• In problems, substitutions or transformations often reveal simpler forms of these functions, aiding in solving complex limits.

Polynomial Functions

Polynomial functions are expressions made up of variables raised to whole number powers, often combined linearly. For example, \( g(x) = x^3 h(x) \) is a polynomial expression if \( h(x) \) itself is polynomial.

These functions are characterized by their smooth curves and dependably predictable rates of change. They are immensely useful in calculus due to their straightforward derivatives and integrals. Comparing polynomial growth to exponential growth, polynomials are much slower and make a reasonable component to counterbalance exponentials in limit problems.

• Polynomials, such as \( x^2 \) or \( x^3 \), effectively counterbalance exponentials in certain limits by growing slower.
• Terms like \( 3x^2 \) in derivatives show how polynomial multiplication helps manage rapid growth rates of other components, ensuring limit existence.
• Polynomial functions are continuous everywhere, which means they do not have disruptions, facilitating easier limit finding.

Continuity

Continuity in a function implies that there are no breaks, jumps, or holes at any given point within its domain. If \( g(x) \) or any of its derivatives are continuous, it means that small changes in \( x \) will result in small changes in \( g(x) \), enhancing the possibility of finding and evaluating limits.

For the given function \( f(x) = g'(x) \, \frac{e^{1/x} - e^{-1/x}}{e^{y_x} + e^{-1/x}} \), having \( g'(x) \) as a continuous function ensures all resulting expressions remain well-behaved as \( x \to 0 \).

• Continuity ensures that limit evaluation is smooth and without surprises.
• When paired with polynomial structures, continuity aids in counter-energy to extreme exponential values.
• For limits, continuity checks if \( \lim_{{x \to a}} f(x) = f(a) \), supporting determinant limit conclusions.