Problem 462 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \text { Find a polynomial } f(x) \text { of least degree such that } \lim _{x \rightarrow 0}\left(2+\frac{f(x)}{x^{2}}\right)^{\frac{1}{x}}=e^{2} \text { . } $$

Step-by-Step Solution

Answer

The polynomial \(f(x)\) is \(4x^{2}\).

Step 1: Simplify the function inside the limit

Rewrite the function to match the formula for \(e\). Divide both sides of the given equation by \(2\). Now, the equation becomes: \(\lim _{x \rightarrow 0}\left(1+\frac{f(x)}{2x^{2}}\right)^{x} = e^{2}\). This now matches the formula for \(e\), and we can conclude that \(\frac{f(x)}{2x^{2}} = 2\).

Step 2: Solve for the polynomial \(f(x)\)

The next step is to isolate \(f(x)\). Multiply both sides of the equation by \(2x^{2}\) to get \(f(x) = 4x^{2}\). Therefore, the polynomial of least degree such that the limit of the given expression as \(x\) approaches \(0\) is \(e^{2}\) is \(f(x) = 4x^{2}\).

Key Concepts

Polynomial functions

Polynomial functions are a fundamental concept in algebra and calculus. A polynomial is an expression made up of variables (like \(x\)) that are raised to whole number exponents and combined using addition, subtraction, and multiplication.
For example, a simple polynomial function can look like \(f(x) = 4x^2 + 3x + 2\). Each term in this expression, such as \(4x^2\) or \(3x\), is a part of the polynomial.

When dealing with limits involving polynomials, it’s important to find the term with the highest power, as this often dominates the behavior as \(x\) approaches infinity or zero. In our exercise, the polynomial \(f(x) = 4x^2\) is crucial because of its direct contribution to forming the limit.

• Polynomial degree: The highest power of \(x\) in the polynomial. Here, it is 2 in \(4x^2\).
• Coefficients: Numerical factors like 4 in \(4x^2\) which affect the stretching or compressing of the polynomial graph.

Using these properties, we identified the simplest polynomial \(f(x) = 4x^2\) that satisfies our limit condition. Understanding polynomials helps in predicting the behavior of functions as they approach specific points.

Exponential functions

Exponential functions are characterized by a constant base raised to a variable exponent, typically written as \(a^x\), where \(a\) is a positive constant. For example, \(e^x\) is a common exponential function, where \(e\) is Euler's number, approximately equal to 2.718.
These functions show rapid growth or decay, depending on whether the exponent is increasing or decreasing. Such behavior is highlighted in our exercise where \(e^{2}\) is the result of evaluating the limit.

• Base \(e\): Unlike constant bases, \(e\) uniquely appears in continuous growth scenarios, such as compound interest and population growth.
• Inverse function: The natural logarithm, \(\ln(x)\), acts as the inverse of \(e^x\).

In this exercise, understanding the properties of exponential functions allowed us to recognize the format of \(e^2\) within the limit expression \(\lim_{x \rightarrow 0}\left(1+\frac{f(x)}{2x^2}\right)^x\), ensuring that our polynomial \(f(x)\) was correctly chosen to match this exponential format.

Limit properties

Limits are a cornerstone in calculus, providing a way to understand the behavior of functions as variables approach a specific value. The limit properties help simplify complex expressions and find values that functions tend to approach.
In our exercise, we utilized a limit property that allows expressions of the form \(\left(1+\frac{g(x)}{h(x)}\right)^{h(x)}\) to approach \(e^{a}\) when \(\frac{g(x)}{h(x)}\) tends to \(a\).
Key properties of limits include:

• Product and quotient properties: These help in breaking down complicated expressions, allowing limits of simpler expressions to be computed individually.
• Exponential and logarithm related limits: These apply to expressions involving \(e\), where limits help convert an exponential approach to a function into a simpler arithmetic expression.

By recognizing the structure required to achieve an \(e\)-form in our task, we simplified and rearranged expressions to identify \(f(x)\). This understanding of limit properties is vital for solving calculus problems efficiently.