Problem 30 - Precalculus: Mathematics for Calculus

Question

One-Sided Limits Graph the piecewise-defined function and use your graph to find the values of the limits, if they exist. $$f(x)=\left\\{\begin{array}{ll} 2 & \text { if } x<0 \\ x+1 & \text { if } x \geq 0 \end{array}\right.$$ (a) \(\lim _{x \rightarrow 0^{-}} f(x)\) (b) \(\lim _{x \rightarrow 0^{+}} f(x)\) (c) \(\lim _{x \rightarrow 0} f(x)\)

Step-by-Step Solution

Answer

(a) 2, (b) 1, (c) Does not exist.

Step 1: Understanding the Piecewise Function

The piecewise function is defined as follows: \(f(x) = 2\) for \(x < 0\) and \(f(x) = x + 1\) for \(x \geq 0\). This means there are two parts we need to understand and analyze for their behaviors near \(x = 0\).

Step 2: Graphing the Function

On a graph, plot two different segments: a horizontal line at \(y = 2\) for all values of \(x < 0\), and a line with a slope of 1 starting at \(y = 1\) when \(x = 0\) (since \(y = x + 1\) implies \(1\) when \(x = 0\)). Mark the point at \(x = 0\) as closed for \(x + 1\) (since \(0 + 1 = 1\)) and open for \(y = 2\).

Step 3: Finding \(\lim _{x \rightarrow 0^{-}} f(x)\)

Evaluate the left-hand limit where \(x\) approaches 0 from the negative side. From the graph, for \(x < 0\), \(f(x) = 2\). Thus, \(\lim _{x \rightarrow 0^{-}} f(x) = 2\).

Step 4: Finding \(\lim _{x \rightarrow 0^{+}} f(x)\)

Evaluate the right-hand limit where \(x\) approaches 0 from the positive side. From the graph, for \(x \geq 0\), \(f(x) = x + 1\) which gives \(f(0) = 0 + 1 = 1\). So, \(\lim _{x \rightarrow 0^{+}} f(x) = 1\).

Step 5: Determining \(\lim _{x \rightarrow 0} f(x)\)

Compare the left-hand and right-hand limits. Since \(\lim _{x \rightarrow 0^{-}} f(x) = 2\) and \(\lim _{x \rightarrow 0^{+}} f(x) = 1\), the two limits are not equal. Therefore, the limit \(\lim _{x \rightarrow 0} f(x)\) does not exist.

Key Concepts

Piecewise-Defined Function

A piecewise-defined function is a type of function in which different formulas are used to define different parts of its domain. These functions are divided into "pieces," each corresponding to a specific interval of the domain.
For example, consider the function given in the exercise:

• For all values of \(x < 0\), the function \(f(x)\) is defined as a constant, \(f(x) = 2\). This part of the function is a horizontal line in the graph.
• For all values of \(x \geq 0\), the function follows another rule, \(f(x) = x + 1\). In this case, the function is linear, with a slope of 1 and a y-intercept of 1.

To fully understand a piecewise-defined function, it's crucial to analyze each piece and consider how they fit together, especially at the boundary of their definitions, such as \(x = 0\) in this case.

Graphical Analysis

Graphical analysis involves plotting the function to visually assess its behavior and properties. This is a powerful tool for understanding piecewise-defined functions and their limits.
For the exercise at hand:

• First plot a flat, horizontal line at \(y = 2\) for all \(x < 0\). Since the function is constant in this section, the graph does not change until \(x = 0\).
• Then, for \(x \geq 0\), plot a line with a slope of 1 starting from \(y = 1\), based on the equation \(f(x) = x + 1\).

At the point \(x = 0\), you should indicate that the point for the equation \(x + 1\) is closed (filled) and for \(y = 2\) it's open (unfilled). This graph helps you identify where the limits might differ as \(x\) approaches 0.

Left-Hand Limit

The left-hand limit, denoted \(\lim_{x \to c^-} f(x)\), evaluates the behavior of \(f(x)\) as \(x\) approaches a point \(c\) from the left-hand side or the negative direction.
For the problem at hand, we look at how \(f(x)\) behaves as \(x\) gets closer to 0 from values less than 0:

• Based on the graph, we see that for \(x < 0\), \(f(x) = 2\). Therefore, as \(x\) approaches 0 from the left, the function remains constant.
• This results in the left-hand limit being \(\lim_{x \rightarrow 0^{-}} f(x) = 2\).

The left-hand limit gives us an important insight into the potential discontinuity at the point, which occurs when the left and right limits don't match.

Right-Hand Limit

Conversely, the right-hand limit, expressed as \(\lim_{x \to c^+} f(x)\), examines the value that \(f(x)\) is approaching as \(x\) comes in from the right of \(c\).
In this case, we observe the behavior of \(f(x)\) as \(x\) nears 0 from values greater than or equal to 0:

• From our function definition, when \(x \geq 0\), \(f(x) = x + 1\). Thus, exactly at \(x = 0\), the function equals 1.
• As a result, the right-hand limit is \(\lim_{x \rightarrow 0^{+}} f(x) = 1\).

Comparing the left and right limits helps determine if \(f(x)\) has an overall limit at \(c\). In this exercise, differing values indicate a discontinuity, implying \(\lim_{x \to 0} f(x)\) does not exist.