Problem 33 - Precalculus: Mathematical for Calculus

Question

Let \(f(x)=\left\\{\begin{array}{ll}x-1 & \text { if } x<2 \\ x^{2}-4 x+6 & \text { if } x \geq 2\end{array}\right.\) (a) Find \(\lim _{x \rightarrow 2} f(x)\) and \(\lim _{x \rightarrow 2^{+}} f(x)\) (b) Does \(\lim _{x \rightarrow 2} f(x)\) exist? (c) Sketch the graph of \(f .\)

Step-by-Step Solution

Answer

(a) \( \lim_{x \rightarrow 2^{-}} f(x) = 1 \), \( \lim_{x \rightarrow 2^{+}} f(x) = 2 \); (b) No, it doesn't exist; (c) Graph has a discontinuity at \( x = 2 \).

Step 1: Evaluate One-Sided Limits

First, calculate \( \lim_{x \rightarrow 2^{-}} f(x) \). For \( x < 2 \), the function is given by \( f(x) = x - 1 \). Thus, \( \lim_{x \rightarrow 2^{-}} f(x) = 2 - 1 = 1 \). Next, calculate \( \lim_{x \rightarrow 2^{+}} f(x) \). For \( x \geq 2 \), the function is \( f(x) = x^2 - 4x + 6 \). Evaluating this limit as \( x \rightarrow 2^+ \), we get \( 2^2 - 4 \cdot 2 + 6 = 4 - 8 + 6 = 2 \).

Step 2: Find Two-Sided Limit

The two-sided limit \( \lim_{x \rightarrow 2} f(x) \) exists only if both one-sided limits are equal. Since we found \( \lim_{x \rightarrow 2^{-}} f(x) = 1 \) and \( \lim_{x \rightarrow 2^{+}} f(x) = 2 \), the limits are not equal. Hence, \( \lim_{x \rightarrow 2} f(x) \) does not exist.

Step 3: Determine Existence of Overall Limit

Since the one-sided limits from Step 1 are not equal, the overall two-sided limit \( \lim_{x \rightarrow 2} f(x) \) does not exist. The function behavior changes at \( x = 2 \).

Step 4: Sketch the Graph of f(x)

To sketch \( f(x) \), consider both pieces of the function definition. For \( x < 2 \), the graph is a line \( y = x - 1 \) with a hole at \( (2, 1) \). For \( x \geq 2 \), the graph is a parabola \( y = x^2 - 4x + 6 \) starting from the point \( (2, 2) \). These segments create a step-like feature in the graph at \( x = 2 \).

Key Concepts

Limits

The concept of limits is crucial in calculus. It refers to the value a function approaches as the input approaches a particular point. For a function \( f(x) \), the limit \( \lim_{x \to c} f(x) \) denotes the value \( f(x) \) gets closer to as \( x \) gets closer to \( c \). Calculating limits helps us understand the behavior of functions near specific points, even if the function isn't explicitly defined there. In the context of piecewise functions, this is particularly important because the behavior of the function may change or "jump" at certain points.

One-Sided Limits

One-sided limits are a type of limit that looks at the behavior of a function as it approaches a point from one side - either from the left or the right. In mathematical terms, the left-hand limit is \( \lim_{x \rightarrow c^{-}} f(x) \), analyzing \( f(x) \) as \( x \) approaches \( c \) from the left, while the right-hand limit is \( \lim_{x \rightarrow c^{+}} f(x) \), for \( x \) approaching \( c \) from the right.

• Think of them as zooming in from either side.
• If these one-sided limits are different, the two-sided limit does not exist.

In the exercise provided, one-sided limits were used to determine the behavior of the piecewise function as \( x \to 2 \). Since \( \lim_{x \rightarrow 2^{-}} f(x) = 1 \) and \( \lim_{x \rightarrow 2^{+}} f(x) = 2 \), the limits are not equal, meaning the two-sided limit \( \lim_{x \rightarrow 2} f(x) \) does not exist.

Graphing Piecewise Functions

Graphing piecewise functions involves understanding how the function behaves in different intervals and depicting those behaviors visually. Each "piece" of the function is defined over a specific interval, and the graph will switch from one piece to the next at the endpoints of these intervals. Here are the steps to graph a piecewise function effectively:

• Identify each part of the piecewise function and its defined interval.
• Graph each piece on its respective interval, ensuring continuity or placing a hole where the function isn't defined.
• Notice jumps or breaks in the graph, which occur where one-piece ends and the other begins.

For the function \( f(x) = \begin{cases} x-1, & x<2 \ x^{2}-4x+6, & x\geq 2 \end{cases} \), the graph involves a linear segment and a parabolic curve. The linear part \( y = x - 1 \) is graphed for \( x < 2 \), and at \( x = 2 \), a hole appears at (2,1) because \( x = 2 \) is not included. The curve \( y = x^{2} - 4x + 6 \) begins at \( x = 2 \) with a starting point at (2,2), representing the right continuous limit at this point. This visual representation helps learners easily recognize how the function transitions and where gaps might exist.