Problem 38 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(f(x)=\sum_{n=0}^{\infty} \frac{x^{n}}{n !}(\log a)^{n}\), then at \(x=0, f(x)\) (A) has no limit (B) is discontinuous (C) is continuous but not differentiable (D) is differentiable

Step-by-Step Solution

Answer

(D) is differentiable

Step 1: Identify the Function

The given function is defined as a power series: \( f(x) = \sum_{n=0}^{\infty} \frac{x^n}{n!} (\log a)^n \). This is similar to the power series expansion of the exponential function \( e^{x} \), where \( x \) is replaced by \( x \log a \).

Step 2: Recognize the Series Type

Notice that the series can be rewritten as: \( f(x) = \sum_{n=0}^{\infty} \frac{(x \log a)^n}{n!} \), which matches the form of the exponential series \( e^{x\log a} \). Using the property of exponents, it simplifies to \( f(x) = a^x \).

Step 3: Evaluate at x = 0

Substitute \( x = 0 \) in \( f(x) = a^x \), giving \( f(0) = a^0 = 1 \). This confirms that the value of \( f(0) \) is 1.

Step 4: Determine Continuity at x = 0

Since \( f(x) = a^x \) is a standard exponential function, it is continuous everywhere, including at \( x = 0 \).

Step 5: Determine Differentiability at x = 0

Exponential functions like \( f(x) = a^x \) are differentiable everywhere. Therefore, \( f(x) \) is differentiable at \( x = 0 \).

Key Concepts

Exponential Function

The exponential function is a fundamental mathematical function, denoted as \(e^x\), where \(e\) is a mathematical constant approximately equal to 2.71828. This function has unique properties, making it crucial in calculus, complex analysis, and other fields of mathematics. One prominent property is its representation through a power series:

• \(e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}\)

This series converges for all real values of \(x\), which makes the exponential function defined and smooth across the entire real number line.
The given series \(f(x) = \sum_{n=0}^{\infty} \frac{x^n}{n!} (\log a)^n\) resembles the exponential function, but where \(x\) is replaced by \(x \log a\). This manipulation translates to another exponential function \(a^x\), showcasing the flexibility and wide applicability of exponential functions in mathematical expressions.

Function Differentiability

Differentiability is a key concept in calculus, referring to a function's ability to have a derivative at any given point. A function is said to be differentiable at a point if it has a well-defined tangent at that point, meaning its rate of change can be precisely calculated.

• The derivative essentially represents the slope of a function at a given point.

For the function \(f(x) = a^x\), where \(a\) is a constant, differentiability is straightforward:

• This function is differentiable for all real numbers.
• The derivative with respect to \(x\) is \(f'(x) = a^x \log a\).

Thus, when evaluating at \(x = 0\), the differentiability of \(a^x\) at this point confirms that \(f(x)\) has a well-defined rate of change, proving it is differentiable there. Differentiability at every point where the function is continuous is a powerful aspect of exponential functions.

Continuity of Functions

Continuity in functions means there are no abrupt jumps, breaks, or holes in the graph of the function. Formally, a function is continuous at a point if the limit of the function as it approaches the point is equal to the function’s value at that point.

• Mathematically put, a function \(f(x)\) is continuous at \(x = c\) if \(\lim_{x \to c} f(x) = f(c)\).

The exponential function \(a^x\), known for its continuous nature, holds true for any positive real number \(a\). In the context of this exercise:

• Since \(f(x) = a^x\), continuity at \(x = 0\) ensures that \(\lim_{x \to 0} a^x = a^0 = 1\).

This characteristic verifies the function's smooth behavior without discontinuities at \(x = 0\). Consistently, the exponential function's continuity across all real numbers showcases its importance in mathematics, providing a steady and uninterrupted progression.