Problem 60 - Precalculus with Limits

Question

In Exercises 55-62, graph the function. Determine the limit (if it exists) by evaluating the corresponding one-sided limits. \\[ \lim_{x \to 1} f(x)\ \textrm{where}\ f(x) = \begin{cases} 2x+1, & \quad x<1\\\ 4-x^2, & \quad x\geq1 \end{cases} \\]

Step-by-Step Solution

Answer

The limit \(\lim_{x \to 1} f(x)\) exists and is equal to 3.

Step 1: Graph the function

First, graph the piecewise function \(f(x)\) which is defined as \(f(x) = 2x+1\) for \(x<1\) and \(f(x) = 4-x^2\) for \(x\geq1\). These two cases form two separate graphs on the same coordinate plane.

Step 2: Calculate the left-hand limit

Using the definition of the function for \(x < 1\), calculate \(\lim_{x \to 1^-} f(x)\), which represents the limit as \(x\) approaches 1 from the left. Substitute 1 into the first case, \(2x + 1\), which yields the left-hand limit as \(2(1) + 1 = 3\).

Step 3: Calculate the right-hand limit

Next, using the second definition of the function for \(x \geq 1\), calculate \(\lim_{x \to 1^+} f(x)\), which is the limit as \(x\) approaches 1 from the right. Substitute 1 into the second case, \(4 - x^2\), which gives the right-hand limit as \(4 - 1^2 = 3\).

Step 4: Check for the existence of the limit

The limit \(\lim_{x \to 1} f(x)\) exists if and only if the left-hand limit equals to the right-hand limit. As both the left-hand limit and the right-hand limit are 3, the limit exists and is equal to 3.

Key Concepts

Graphing Piecewise Functions

Graphing piecewise functions is a critical skill in understanding the behavior of functions that have different rules for different parts of their domain. Let's begin with pinpointing what a piecewise function is. Essentially, it is a function comprised of multiple segments or 'pieces', each with its own formula, defined on specific intervals.

When graphing a piecewise function, like the given function \(f(x)\), you have to graph each piece separately. For \(x<1\), plot \(2x+1\) which will be a straight line. For \(x\geq1\), graph \(4-x^2\), which is a downward-opening parabola. Since the functions handle different ranges, they should align properly on the graph to illustrate a coherent picture of \(f(x)\).

Attention to Detail

Make sure to take note of open and closed circles, which indicate whether a particular point is included or excluded from a graph's segment. Open circles represent an excluded point (typically found where functions shift from one rule to another), while closed circles represent an included point. In our exercise, the transition at \(x=1\) will be significant in identifying the correct limits and ensuring a seamless graph.

One-sided Limits

One-sided limits solve the mystery of a function's behavior as you approach a particular point from one direction only—either from the left (\(-\)) or from the right (\(+\)). The notation \(x \to c^-\) represents approaching the point \(c\) from the left, while \(x \to c^+\) signifies approaching \(c\) from the right.

Importantly, one-sided limits are vital in piecewise functions as they can vastly differ on either side of a point. In our exercise, calculating the one-sided limits of \(f(x)\) at \(x=1\) helps establish the definitive limit of the function at that point. If these one-sided limits do not match, the general limit at that point does not exist. The fact that the left-hand limit and the right-hand limit both equal 3 for our function points to a consistent behavior across the defined 'pieces,' leading to the existence of a limit at that point.

Evaluating Limits

Evaluating limits is the process of finding out what value a function approaches as the input approaches a specific value. This 'approach' aspect is crucial; we're not calculating what the function equals at that point, but rather where it's heading as it gets infinitely close to it. When dealing with piecewise functions, as in our exercise, we have to carefully evaluate limits for each piece based on where they're valid.

For our function \(f(x)\), we smartly break the limit evaluation into two one-sided limits, focusing on the approach from the left (\(x \to 1^-\)) and from the right (\(x \to 1^+\)). Evaluating each limit based on their respective function piece ensures precision. Because both one-sided limits agree at 3, we discern that the overall limit as \(x\) approaches 1 is also 3, denoting smooth continuity at that junction match.

A tip for robust understanding: Always check if the function's pieces align at common boundary points when evaluating limits, as this will affect whether or not the overall limit exists.