Problem 47 - A Graphical Approach to Precalculus with Limits

Question

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value. \(\lim _{x \rightarrow 1} f(x),\) where \(f(x)=\left\\{\begin{array}{ll}e^{x} & \text { if } x \leq 1 \\ \sqrt{x} & \text { if } x>1\end{array}\right.\)

Step-by-Step Solution

Answer

The limit does not exist as the one-sided limits are not equal.

Step 1: Understand the function definition

The function \( f(x) \) is a piecewise function defined as \( e^x \) for \( x \leq 1 \) and \( \sqrt{x} \) for \( x > 1 \). We will need to analyze the behavior of both parts of the function as \( x \) approaches 1.

Step 2: Analyze the limit from the left

As \( x \to 1^- \), \( f(x) = e^x \) since this part of the function is defined for \( x \leq 1 \). The limit of \( e^x \) as \( x \to 1 \) is just \( e^1 \), which equals \( e \).

Step 3: Analyze the limit from the right

As \( x \to 1^+ \), \( f(x) = \sqrt{x} \) since this part of the function is defined for \( x > 1 \). The limit of \( \sqrt{x} \) as \( x \to 1 \) is \( \sqrt{1} = 1 \).

Step 4: Determine if the two one-sided limits are equal

For the limit \( \lim_{{x \to 1}} f(x) \) to exist, the left-hand limit (\( e \)) must equal the right-hand limit (\( 1 \)). Since \( e eq 1 \), the two one-sided limits are not equal.

Step 5: Conclusion regarding the limit

Since the left-hand and right-hand limits are not equal, the overall limit \( \lim_{{x \to 1}} f(x) \) does not exist.

Key Concepts

Piecewise Functions

In mathematics, piecewise functions are made up of several sub-functions, each of which applies to a specific part of the domain. This allows us to describe situations where a function behaves differently in different intervals or under different conditions. For the exercise given, the piecewise function is defined as follows:

• For values of \( x \) less than or equal to 1, the function is described by \( e^x \).
• For values of \( x \) greater than 1, the function is described by \( \sqrt{x} \).

Understanding piecewise functions requires attention to these conditions to correctly apply the appropriate expression. This becomes crucial when dealing with limits, as each segment must be evaluated within the context of its own domain. Knowing which sub-function to apply is key to solving limits involving piecewise functions.

One-Sided Limits

One-sided limits help us understand how a function behaves as the input approaches a particular point from one side only, either from the left or the right. In this exercise, we analyze the behavior of the function \( f(x) \) as \( x \) approaches 1. Here’s how:

• The left-hand limit (approaching from values less than 1) was calculated using \( e^x \) because for \( x \leq 1 \), \( f(x) = e^x \). This results in a limit of \( e \).
• The right-hand limit (coming from values greater than 1) was calculated using \( \sqrt{x} \) since for \( x > 1 \), \( f(x) = \sqrt{x} \). This results in a limit of 1.

To check if the overall limit exists at that point, both the left-hand and right-hand limits must be equal. Since \( e \) is not equal to 1, the limits from each side do not match, indicating that the overall limit does not exist at \( x = 1 \). One-sided limits are a critical tool in determining discontinuities or seeing where a fuller limit may fail to exist.

Graphical Analysis

Graphical analysis provides a visual representation of how a function behaves, making it easier to understand the behavior of limits and continuity. For the piecewise function in this exercise, imagine plotting parts of the function on a graph:

• Plotting \( e^x \) for \( x \leq 1 \) typically forms part of the exponential curve, smoothly growing upward.
• For \( x > 1 \), plotting \( \sqrt{x} \) involves a curve starting from \( x = 1 \), creating a different trajectory.

By observing these plots, one can see a clear graphical "jump" at \( x = 1 \). This jump illustrates why the overall limit at \( x = 1 \) does not exist: the left-hand limit rises to \( e \), whereas the right-hand limit sits at 1, differing significantly. Graphical analysis not only confirms the numerical findings but also enhances understanding by giving an intuitive view of the function's behavior. Such visual aids are invaluable in explaining complex concepts like limits and continuity in piecewise functions.