Problem 1 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Given \(\begin{aligned} f(x) &=(1+x)^{\frac{1}{x}}, \quad x \neq 0 \\ &=e, \quad x=0 \end{aligned}\)

Step-by-Step Solution

Answer

The given piecewise function is defined as \(f(x) = (1+x)^{\frac{1}{x}}\) for \(x \neq 0\) and \(f(x) = e\) for \(x = 0\). We find that the limit as \(x\) approaches 0 of \((1+x)^{\frac{1}{x}}\) is e, making the function continuous at x=0 and for all x. Additionally, the function is always positive.

Step 1: Understand the given function

Consider the given piecewise function: \[f(x) = \begin{cases} (1+x)^{\frac{1}{x}}, & \text{if } x \neq 0; \\ e, & \text{if } x=0. \end{cases}\] When x is not equal to 0, the function is defined as \((1+x)^{\frac{1}{x}}\), and when x = 0, the function takes the value of e, which is approximately 2.718.

Step 2: Explain the importance of the limit

Since the function is defined in a piecewise way, we need to explain why it takes the value e when x = 0. To do this, we will examine the limit of the function as x approaches 0: \[\lim_{x \to 0} (1+x)^{\frac{1}{x}}\]

Step 3: Evaluate the limit

In order to evaluate the above limit, we need to recall a fundamental limit, which states that: \[\lim_{x \to 0} (1+x)^{\frac{1}{x}} = e\] Now, since we have proved that the limit of the function as x approaches 0 is e, we can say that the value of the function at x = 0 is e, and therefore the function is continuous at x = 0.

Step 4: Explain the properties of the function

As we have found the limit at x=0, it is important to note that the function is continuous for all x, as it is a composition of continuous functions when x is not 0 and has a defined value at x = 0 based on the limit. It is also clear that the function is always positive, as \((1+x)^{\frac{1}{x}}\) is positive for all x ≠ 0 and the value e (approximately 2.718) is positive as well. In conclusion, the given function is a continuous function defined in a piecewise manner, with the value e at x = 0 based on the fundamental limit. The function is also always positive for any x.

Key Concepts

Limits in Calculus

The concept of limits is a foundational element in calculus that helps us understand the behavior of functions as their inputs approach some value. In this particular exercise, taking the limit is crucial to determine the value of the piecewise function at the discontinuity point, which is at \(x = 0\).
For the function \(f(x) = (1+x)^{\frac{1}{x}}\), the challenge is to understand what happens as \(x\) gets infinitely close to zero but does not actually reach it. By evaluating the limit \[ \lim_{x \to 0} (1+x)^{\frac{1}{x}} \]we find that the expression approaches \(e\), Euler's number, which is approximately 2.718. This fundamental limit is critical because it confirms that the function approaches a specific, finite value even though the input \(x\) is near a point where the function is not directly defined.
Thus, limits not only allow us to assign a sensible value to the function at \(x=0\), but they also ensure continuity at this point, enabling a complete understanding of the function's behavior across its domain.

Piecewise Functions

Piecewise functions are mathematical expressions defined by multiple sub-functions, each applying to a specific interval of the function's domain. In this exercise, the function \(f(x)\) is defined piecewise as follows:

• \((1+x)^{\frac{1}{x}}\) when \(x eq 0\)
• \(e\) when \(x = 0\)

The reason a piecewise function is used here is because the expression \((1+x)^{\frac{1}{x}}\) becomes indeterminate when \(x=0\). However, by assigning the limit value \(e\) at \(0\), which is consistent with the limit calculation, we ensure the function is continuous.
Continuity is crucial because it ensures no jumps, gaps, or holes in the function graph, making it more predictable and manageable. Understanding how piecewise functions operate allows us to model complex scenarios more accurately, which are common in real-world applications involving sudden changes or different conditions.

Exponential Function

Exponential functions are a key type of function in mathematics with a variable in the exponent, as seen in the expression \((1+x)^{\frac{1}{x}}\). These functions have unique properties, such as rapid growth or decay, depending significantly on the base of the exponent.
In the context of this exercise, the expression can be viewed as an exponential form with a complex variable exponent. An interesting feature occurs as \(x\) approaches zero, where the expression aligns with the definition of the mathematical constant \(e\).
Exponential functions appear frequently in various scientific fields, such as physics, biology, and economics, each time taking advantage of their properties of change and scaling. Understanding the behavior of exponential-based expressions like \((1+x)^{\frac{1}{x}}\) can lead to deeper insights in these areas, as their characteristic properties describe natural phenomena occurring around us.