Problem 28 - A Graphical Approach to Precalculus with Limits

Question

Determine each limit. \(f(x)=\left\\{\begin{array}{ll}2 x+3 & \text { if } x<1 \\ 4 & \text { if } x=1 \\ x^{2} & \text { if } x>1\end{array}\right.\) (a) \(\lim _{x \rightarrow 1^{+}} f(x)\) (b) \(\lim _{x \rightarrow 1^{-}} f(x)\)

Step-by-Step Solution

Answer

(a) 1, (b) 5

Step 1: Understand Right-Hand Limit

To find the right-hand limit, we consider values of \(x\) approaching 1 from the right (greater than 1). The function definition for \(x > 1\) is \(f(x) = x^2\).

Step 2: Evaluate Right-Hand Limit

The right-hand limit \(\lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} x^2\). As \(x\) gets very close to 1 from the right, \(x^2\) gets very close to \(1^2 = 1\). Thus, \(\lim_{x \to 1^+} f(x) = 1\).

Step 3: Understand Left-Hand Limit

To find the left-hand limit, we look at values of \(x\) approaching 1 from the left (less than 1). For \(x < 1\), the function is defined as \(f(x) = 2x + 3\).

Step 4: Evaluate Left-Hand Limit

The left-hand limit \(\lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} (2x + 3)\). As \(x\) approaches 1 from the left, the expression \(2x + 3\) approaches \(2 \times 1 + 3 = 5\). Therefore, \(\lim_{x \to 1^-} f(x) = 5\).

Key Concepts

Right-Hand Limit

When we talk about the right-hand limit in calculus, we're referring to the behavior of a function as the input values approach a specific point from the right. This means we are considering values of the variable that are slightly more than the particular point. It's like inching towards the point just above it on the number line.

For the given function, we want to evaluate the limit as \(x\) approaches 1 from the right. This is written as \(\lim_{x \to 1^+} f(x)\). Here, we're interested in the section of our piecewise function where \(x > 1\). According to the function's definition, this part will use the equation \(f(x) = x^2\).

• Evaluate \(\lim_{x \to 1^+} x^2\): As \(x\) gets closer and closer to 1 from the right, the value of \(x^2\) will get closer and closer to \(1^2 = 1\).

Thus, the right-hand limit of the function, as \(x\) approaches 1 from values greater than 1, is 1.

Left-Hand Limit

The left-hand limit focuses on how a function behaves as we approach a particular point from the left side. In other words, it observes the behavior of the function as the input value decreases towards the particular point but does not exceed it.

In our exercise, we want to understand the limit as \(x\) nears 1 from the left. This is noted as \(\lim_{x \to 1^-} f(x)\). For this, we should consider the part of the piecewise function where \(x < 1\), which in this case is defined as \(f(x) = 2x + 3\).

• Evaluate \(\lim_{x \to 1^-} (2x + 3)\): As \(x\) comes closer to 1 from the left, the expression \(2x + 3\) approaches \(2 \times 1 + 3 = 5\).

This means the left-hand limit of the function, as \(x\) gets near 1 from values smaller than 1, equals 5.

Piecewise Functions

Piecewise functions can seem a bit tricky at first, but they're just functions that have different rules or expressions depending on the input value. Think of it like a set of instructions: "If x is less than this, do that; if x is more than this, do something else."

In the case of the function provided, it can be visualized as having three distinct parts:

• For \(x < 1\), the function follows the expression \(f(x) = 2x + 3\).
• If \(x = 1\), then \(f(x) = 4\), a single point on the graph.
• For \(x > 1\), the function takes the form \(f(x) = x^2\).

This is useful for determining limits because depending on the direction from which \(x\) approaches a point, you will use different sections of the function. Understanding which piece of the piecewise function to use is key to correctly finding limits and ensures accurate interpretations of the behavior of the function around certain points.