Problem 18 - APEX Calculus

Question

Evaluate the given limits of the piecewise defined functions \(f\). \(f(x)=\left\\{\begin{array}{cc}x+1 & x<1 \\ 1 & x=1 \\ x-1 & x>1\end{array}\right.\) (a) \(\lim _{x \rightarrow 1^{-}} f(x)\) (c) \(\lim _{x \rightarrow 1} f(x)\) (b) \(\lim _{x \rightarrow 1^{+}} f(x)\) (d) \(f(1)\)

Step-by-Step Solution

Answer

(a) 2, (b) 0, (c) does not exist, (d) 1.

Step 1: Evaluate the Left-Hand Limit

The left-hand limit as \(x\) approaches 1 is denoted as \(\lim _{x \rightarrow 1^{-}} f(x)\). For \(x<1\), the function \(f(x) = x + 1\). So, \(\lim _{x \rightarrow 1^{-}} f(x) = 1 + 1 = 2\).

Step 2: Evaluate the Right-Hand Limit

The right-hand limit as \(x\) approaches 1 is represented as \(\lim _{x \rightarrow 1^{+}} f(x)\). For \(x>1\), the function \(f(x) = x - 1\). Thus, \(\lim _{x \rightarrow 1^{+}} f(x) = 1 - 1 = 0\).

Step 3: Evaluate the Two-Sided Limit

The two-sided limit \(\lim _{x \rightarrow 1} f(x)\) exists only if both the left-hand limit and the right-hand limit are equal. Since \(\lim _{x \rightarrow 1^{-}} f(x) = 2\) and \(\lim _{x \rightarrow 1^{+}} f(x) = 0\), the two-sided limit does not exist.

Step 4: Determine the Value of the Function at the Point

The value of the function \(f\) at \(x = 1\) is given directly by the piecewise function: \(f(1) = 1\).

Key Concepts

Piecewise Functions

Piecewise functions are mathematical functions that have different expressions or rules for different parts of their domain. They can be a bit tricky at first, but they are quite common in practical scenarios. For example, a taxi fare might cost a flat fee for the first mile and a different rate for miles beyond that. In the given exercise, we have:

• For values of \(x\) less than 1, the function follows \(f(x) = x + 1\).
• For \(x = 1\), the function value is simply 1.
• For values greater than 1, the function follows \(f(x) = x - 1\).

To fully grasp a piecewise function, it helps to sketch out or visualize the different segments and how they connect across the different regions of \(x\). This type of function is crucial in understanding the behavior around points where the rules change, such as around \(x = 1\) in this example.

Left-Hand Limit

The concept of a left-hand limit refers to the behavior of a function as the variable approaches a particular value from the left side, or negative direction. In mathematical notation, the left-hand limit as \(x\) approaches \(c\) is written as \(\lim_{x \to c^{-}} f(x)\). This means we only consider values of \(x\) that are less than \(c\).

In our exercise, to find \(\lim_{x \to 1^{-}} f(x)\) for the piecewise function given, we look at the rule that applies for \(x < 1\), which is \(f(x) = x + 1\). Since we are approaching 1 from the left, we substitute \(x = 1\) into \(x + 1\), which results in 2. Hence, \(\lim_{x \to 1^{-}} f(x) = 2\). This helps us gauge the function's behavior just before it hits \(x = 1\).

Right-Hand Limit

The right-hand limit examines what happens as a function approaches a certain point from the right side, or positive direction. It's denoted by \(\lim_{x \to c^{+}} f(x)\). This concept is crucial when dealing with piecewise functions, as the behavior can be quite different when approaching from either side.

• For our exercise - as \(x\) approaches 1 from the right (\(x > 1\)), the applicable rule is \(f(x) = x - 1\).
• Substituting \(x = 1\) in this expression gives us 0.
• Therefore, the right-hand limit \(\lim_{x \to 1^{+}} f(x) = 0\).

Understanding both left-hand and right-hand limits helps us determine if there is a jump, hole, or no discontinuity at the point in question.

Two-Sided Limit

A two-sided limit exists at a particular point if the left-hand limit and the right-hand limit at that point are equal. This means the function approaches the same value regardless of the direction from which \(x\) approaches. It is represented by \(\lim_{x \to c} f(x)\).

In the exercise, to find \(\lim_{x \to 1} f(x)\), both the left-hand and right-hand limits are examined:

• Left-hand limit (approaching 1 from the left) is 2: \(\lim_{x \to 1^{-}} f(x) = 2\).
• Right-hand limit (approaching 1 from the right) is 0: \(\lim_{x \to 1^{+}} f(x) = 0\).

Since these are not equal, the two-sided limit \(\lim_{x \to 1} f(x)\) does not exist at \(x = 1\). This indicates there is a discontinuity at this point, which can often be visualized in a graph as a jump or a hole at \(x = 1\).