Problem 21 - APEX Calculus

Question

Evaluate the given limits of the piecewise defined functions \(f\). \(\begin{array}{ll}f(x)=\left\\{\begin{array}{cc}\frac{|x|}{x} & x \neq 0 \\ 0 & x=0\end{array}\right. & \\ \begin{array}{ll}\text { (a) } \lim _{x \rightarrow 0^{-}} f(x) & \text { (c) } \lim _{x \rightarrow 0} f(x) \\ \text { (b) } \lim _{x \rightarrow 0^{+}} f(x) & \text { (d) } f(0)\end{array}\end{array}\)

Step-by-Step Solution

Answer

(a) -1, (b) 1, (c) Does not exist, (d) 0.

Step 1: Evaluate \( \lim_{x \to 0^-} f(x) \)

For \( x \to 0^- \), \( x \) approaches zero from the left (negative values). Thus, \( f(x) = \frac{|x|}{x} \) becomes \( f(x) = \frac{-x}{x} = -1 \) when \( x < 0 \). Therefore, \( \lim_{x \to 0^-} f(x) = -1 \).

Step 2: Evaluate \( \lim_{x \to 0^+} f(x) \)

For \( x \to 0^+ \), \( x \) approaches zero from the right (positive values). Thus, \( f(x) = \frac{|x|}{x} \) becomes \( f(x) = \frac{x}{x} = 1 \) when \( x > 0 \). Therefore, \( \lim_{x \to 0^+} f(x) = 1 \).

Step 3: Evaluate \( \lim_{x \to 0} f(x) \)

The limit \( \lim_{x \to 0} f(x) \) exists only if \( \lim_{x \to 0^-} f(x) \) and \( \lim_{x \to 0^+} f(x) \) are equal. Since \( \lim_{x \to 0^-} f(x) = -1 eq 1 = \lim_{x \to 0^+} f(x) \), the limit does not exist.

Step 4: Evaluate \( f(0) \)

The value \( f(0) \) is defined in the piecewise function as 0 when \( x = 0 \). Thus, \( f(0) = 0 \).

Key Concepts

Limits

Limits are a fundamental concept in calculus, serving as the foundation for topics like derivatives and integrals. A limit describes the value that a function approaches as the input (or independent variable) gets arbitrarily close to a particular point. Understanding limits helps with analyzing the behavior of functions at certain points, which is especially crucial when a function is not explicitly defined at those points.

For the function we are evaluating, considering the limit as it approaches zero from both sides gives insight into its behavior near that critical point. Limits are a gateway to understanding continuous and discontinuous behavior in functions, as they define how a function responds as it nears a point where it might not be initially clear what the function's value is.

Piecewise Functions

Piecewise functions are defined by different expressions for different intervals of the input variable. They allow us to describe functions that have multiple behaviors based on the input. In the given function,

• For any non-zero value of x, the function is \( f(x) = \frac{|x|}{x} \)
• At x = 0, \( f(x) = 0 \)

Such a structure is essentially useful for modeling situations where a behavior changes at a certain point. Understanding piecewise functions is key to analyzing how different rules apply in different regions, making them crucial in engineering and various applied sciences. Piecewise definitions are often utilized to maintain realistic representations of mechanical and physical properties that change distinctly under varied conditions.

Left-Hand Limit

The left-hand limit is concerned with the value a function approaches as the independent variable approaches a specific point from the left. It is denoted as \( \lim_{x \to c^-} f(x) \). This emphasizes focusing purely on the values of \( x \), which are less than \( c \) as those values approach \( c \).

In our example, \( \lim_{x \to 0^-} f(x) = -1 \), since approaching zero from negative values correctly reads the function as \( \frac{-x}{x} = -1 \). Understanding the left-hand limit is critical for analyzing situations where you need to know what happens before an abrupt change at a specific point, crucial for determining the continuity and limits at those points.

Right-Hand Limit

A right-hand limit focus on the value a function approaches as the independent variable approaches a specific point from the right. It is denoted as \( \lim_{x \to c^+} f(x) \). Here, we analyze values greater than \( c \).

For \( x \to 0^+ \), the expression simplifies to \( \lim_{x \to 0^+} f(x) = 1 \) since the values come from positive numbers and the function turns into \( \frac{x}{x} = 1 \). Right-hand limits help in evaluating functions around points of interest from the positive direction, and understanding these can reveal much about functions that behave differently on opposing sides of a critical point.

Function Evaluation

Function evaluation is the process of determining the specific output of a function given an input. It generally involves substituting the input value into the function's performance to obtain the result. In the context of piecewise functions, you must carefully choose the appropriate expression based on the value of the input.

In this particular problem, \( f(0) = 0 \) because the function's piecewise definition assigns a value of 0 specifically when \( x = 0 \). Function evaluation ensures you obtain the correct result unusual points where other branches of the function might not apply, paving way for analyzing discontinuities and sudden changes.