Problem 38 - Contemporary Precalculus

Question

Show that \(f(x)=\sqrt{|x|}\) is continuous at \(x=0\)

Step-by-Step Solution

Answer

Explain your answer using a step-by-step solution. #Answer# Yes, the function \(f(x) = \sqrt{|x|}\) is continuous at \(x=0\). This can be proven using the following step-by-step solution: 1. Find the left limit of the function as x approaches 0: \(\lim_{x \to 0^-} \sqrt{-x} = 0\). 2. Find the right limit of the function as x approaches 0: \(\lim_{x \to 0^+} \sqrt{x} = 0\). 3. Compare the limits and check for continuity: Since \(\lim_{x \to 0^-} f(x) = f(0) = \lim_{x \to 0^+} f(x)\), the function \(f(x) = \sqrt{|x|}\) is continuous at \(x=0\).

Step 1: Find the left limit of the function as x approaches 0

To find the left limit \(\lim_{x \to 0^-} f(x)\), we analyze the function when \(x < 0\) on the left side of \(x = 0\). It's helpful to first rewrite the function \(f(x) = \sqrt{|x|}\) as follows: $$ f(x) = \begin{cases} \sqrt{-x} & \text{for } x < 0 \\ \sqrt{x} & \text{for } x \ge 0 \end{cases} $$ As we're looking for the left limit, we can focus on the case when x < 0 and calculate the limit as x approaches 0 from the left side: $$\lim_{x \to 0^-} \sqrt{-x}$$ Since the square root of any real number is always non-negative and \(0^- < 0\), the limit is: $$\lim_{x \to 0^-} \sqrt{-x} = 0$$

Step 2: Find the right limit of the function as x approaches 0

To find the right limit \(\lim_{x \to 0^+} f(x)\), we analyze the function when \(x \ge 0\) on the right side of \(x = 0\): $$\lim_{x \to 0^+} \sqrt{x}$$ Since the square root of any real number is always non-negative and \(0^+ \ge 0\), the limit is: $$\lim_{x \to 0^+} \sqrt{x} = 0$$

Step 3: Compare the limits and check for continuity

Now that we have found both left and right limits, we can compare them to the value of the function at \(x=0\): $$f(0) = \sqrt{|0|} = \sqrt{0} = 0$$ Since \(\lim_{x \to 0^-} f(x) = f(0) = \lim_{x \to 0^+} f(x)\), we can conclude that the function \(f(x) = \sqrt{|x|}\) is continuous at \(x=0\).

Key Concepts

Limits of Functions

Understanding the limits of functions is essential when learning calculus, as it lays the foundation for concepts like continuity, derivatives, and integrals. A limit describes the behavior of a function as the input approaches a certain value. We say that the limit of function f(x) as x approaches a value c is L, denoted by \(\lim_{x \to c} f(x) = L\), if we can make the values of f(x) as close as desired to L by taking x sufficiently close to c.

The beauty of limits is that they help us handle situations where the function is approaching a particular value, but does not necessarily ever reach it. This occurs often with functions that have a point of discontinuity or when we're investigating the function's behavior at infinity.

In our exercise example, we considered both the left and right limits as x approaches zero. When both sides tend toward the same value, and the function actually equals that value when x is zero, then the function is continuous at that point.

Piecewise Functions

Piecewise functions are like mathematical chameleons, adapting their formula depending on the value of x. They are defined by different expressions for different intervals of x. This is particularly useful when a function behaves differently in separate parts of its domain.

For example, the function \(f(x) = \sqrt{|x|}\) is a piecewise function because the definition of square root changes based on whether x is non-negative (\(x \ge 0\)) or negative (\(x < 0\)). As a result, this function combines two 'pieces', one for \(\sqrt{x}\) when \(x \ge 0\) and another for \(\sqrt{-x}\) when \(x < 0\).

When evaluating piecewise functions for continuity, we look at the limits of each 'piece' at the points of change. To be continuous, the limits from both directions must agree, and the function's value must match this limit at the point.

Absolute Value Properties

The absolute value function, represented by \( |x| \), gives the distance between the number x and zero on a number line, disregarding the direction. This means negative numbers are made positive, as distance is always non-negative.

A crucial property is that \(|x|\) is equal to x if x is greater than or equal to zero, and -x if x is less than zero. This property helps us break down the function \(f(x) = \sqrt{|x|}\) into a piecewise function, simplifying the process of finding limits for values around zero.

Utilizing these properties is essential when evaluating functions involving absolute values for continuity. When we examine the limits as x approaches a point from either side, the absolute value helps ensure that both sides are addressed correctly, and the function’s continuity can be accurately determined.