Question

Find the limits of the following: . If \(f(x)=\left\\{\begin{array}{cl}e^{x} & \text { for } \quad 0 \leq x<1 \\\ x^{2} e^{x} & \text { for } \quad 1 \leq x \leq 5\end{array}\right.\), find \(\lim _{x \rightarrow 1} f(x)\)

Short answer

Answer: The limit as \(x\) approaches 1 for the function \(f(x)\) is \(\mathbf{e}\).

Step-by-step solution

Determine the appropriate function

As x approaches 1, we will use the function \(x^2e^x\) because it is defined for \(1 \leq x \leq 5\).

Evaluate the limit of the selected function

Now, let's compute the limit: \(\lim_{x \to 1} x^2e^x\). Since the function \(x^2e^x\) is continuous and well-defined at x=1, we can simply find the limit by plugging in \(x=1\): \(\lim_{x \to 1} x^2e^x = (1^2)e^1 = e\) So, the limit as \(x\) approaches 1 for the function \(f(x)\) is \(\lim_{x \rightarrow 1} f(x) = e\).

Key concepts

Limits and Continuity

Understanding the behavior of mathematical functions as they approach a certain value is a foundational concept in calculus. This is where the concept of limits comes into play. A limit captures the value that a function approaches as the input (usually denoted as \( x \)) gets closer and closer to a specific point. In the exercise provided, we investigate the limit of a piecewise function as \( x \) approaches 1.

Limits are deeply connected to the idea of continuity. A function is continuous at a point if the limit as \( x \) approaches that point is equal to the function's value at that point. If a function is not continuous, it can have jumps, holes, or asymptotes - which are essential considerations when finding limits. The function in our exercise, \( f(x) \), is continuous at \( x = 1 \), as seen by the seamless transition between the piecewise components. To demonstrate this, we calculate the limit using the value of \( x \) at the point where it is defined, resulting in the answer \( e \), thereby confirming the continuity of the function at that point.

Piecewise Functions

Piecewise functions are functions that have different expressions for different intervals of the domain. In the original exercise, \( f(x) \) is defined by two expressions, creating a piecewise function: one involving an exponential function for values of \( x \) from 0 up to but not including 1, and another involving both a polynomial and an exponential function for values of \( x \) including 1 up to 5.

To find the limit of a piecewise function as \( x \) approaches a point where there might be a crossover, we need to consider the appropriate expression for that point. For limits involving a piecewise function like \( f(x) \) as \( x \) approaches 1, it is vital to select the correct function expression that corresponds to the value 1. In this case, we use \( x^2e^x \) because it is the expression defined at \( x = 1 \). If there had been different expressions on either side of 1, the limit may have been more complex to evaluate, possibly involving one-sided limits.

Exponential Functions Calculus

Exponential functions in calculus are functions of the form \( a^x \), where \( a \) is a positive constant. One of the most important and common bases is the natural base \( e \), which is approximately equal to 2.71828. The function \( e^x \) appears in various natural processes and is distinguished by the fact that its rate of growth is directly proportional to its current value, commonly known as exponential growth.

In the realm of calculus, the exponential function \( e^x \) plays a significant role due to its unique properties. One such property is that the derivative of \( e^x \) with respect to \( x \) is simply \( e^x \) again. This implies that exponential functions are continuously differentiable, and they are their own derivatives, which makes solving calculus problems involving \( e^x \) often more straightforward. Regarding limits, this also means that exponential functions are continuous for all real numbers, which simplifies calculations of limits, such as the one exemplified in the exercise where we find the limit of \( x^2e^x \) as \( x \) approaches 1 to be simply \( e \).