Question

Show that \(\lim _{x \rightarrow 0}|x|=0\) by first evaluating \(\lim _{x \rightarrow 0^{-}}|x|\) and \(\lim _{x \rightarrow 0^{+}}|x| .\) Recall that $$ |x|=\left\\{\begin{array}{ll} x & \text { if } x \geq 0 \\ -x & \text { if } x<0 \end{array}\right. $$.

Short answer

Answer: The limit of the absolute value function as x approaches 0 is 0.

Step-by-step solution

Evaluate the limit as x approaches 0 from the left side

For \(x < 0\), the absolute value function is defined as \(|x| = -x\). To find the limit as \(x \rightarrow 0^-\), let's evaluate the function when \(x\) is approaching \(0\) from the left side: $$ \lim _{x \rightarrow 0^{-}}|x|=\lim _{x \rightarrow 0^{-}}(-x) $$ Since the function is just \(-x\), the limit is simply the value of \(-x\) as \(x\) approaches \(0\). As \(x\) gets closer and closer to \(0\) from the left side, \(-x\) gets closer and closer to \(0\) as well. Therefore, the limit is \(0\).

Evaluate the limit as x approaches 0 from the right side

For \(x \geq 0\), the absolute value function is defined as \(|x| = x\). To find the limit as \(x \rightarrow 0^+\), let's evaluate the function when \(x\) is approaching \(0\) from the right side: $$ \lim _{x \rightarrow 0^{+}}|x|=\lim _{x \rightarrow 0^{+}}(x) $$ Once again, since the function is just \(x\), the limit is simply the value of \(x\) as it approaches \(0\). As \(x\) gets closer and closer to \(0\) from the right side, \(x\) also gets closer and closer to \(0\). Therefore, the limit is \(0\).

Combine the limits

Now that we've found the limits from both sides, let's compare their values. We have the following limits as \(x\) approaches \(0\): $$ \lim _{x \rightarrow 0^{-}}|x|=0 \\ \lim _{x \rightarrow 0^{+}}|x|=0 $$ Since the left-hand and right-hand limits are both equal to \(0\), we can conclude that the overall limit of the absolute value function as \(x \rightarrow 0\) is also \(0\): $$ \lim_{x \rightarrow 0}|x| = 0 $$

Key concepts

Limits in Calculus

Understanding limits is crucial for calculus as they help in describing the behavior of functions as inputs approach a particular value. A limit answers the question: What is the value that a function approaches as the input approaches a certain point? For instance, when we discuss the limit of a function as \( x \rightarrow c \), we're considering what value the function approaches as \( x \) gets closer to \( c \). This doesn't necessarily mean \( x \) will reach \( c \), or that the function's value will actually be achieved when \( x = c \); it's about the journey, not the destination.

Limits can be one-sided, meaning they can describe the behavior from only one direction—left or right. In the given exercise, we explored this by looking at the limits as \( x \) approaches zero from both negative and positive directions. When the limit from the left equals the limit from the right, we can assert that the general limit exists and equals that common value. In practice, evaluating limits includes applying various strategies such as factoring, rationalizing, or using special limit laws, to find the limit value without necessarily simplifying the function's expression at the exact point.

Absolute Value Properties

The absolute value of a number is its distance from zero on the number line, regardless of the direction. This non-negative value is denoted as \( |x| \), and has two essential properties:

• \( |x| \) equals \( x \) if \( x \) is greater than or equal to zero.
• \( |x| \) equals \( -x \) if \( x \) is less than zero.

These properties make the absolute value function a piecewise function, as shown in the exercise. In calculus, when dealing with limits involving the absolute value function, it's crucial to look at the behavior from both sides of the point of interest because the function definition changes based on the input value's sign. As seen in our example, we analyzed the limit as the input approaches zero from both negative and positive directions separately, applying the appropriate absolute value definition in each case. Proper understanding and application of absolute value properties allow for accurate limit evaluation of such functions.

One-Sided Limits

One-sided limits specifically refer to the value that a function approaches as the input approaches a certain point from one side—either from the left (denoted as \( x \rightarrow c^- \)) or from the right (denoted as \( x \rightarrow c^+ \)). These are particularly useful for functions that behave differently on either side of a point, which is frequent in piecewise and absolute value functions.

As illustrated in the solved exercise, we separately evaluated the limit of \( |x| \) as \( x \) approached zero from the left and from the right. This distinction is critical because the definition of the absolute value function depends on whether the input is negative or positive. By determining the one-sided limits are both 0, we confirmed that the function has the same behavior approaching zero from either direction. Thus, the overall limit is established. Recognizing when to use one-sided limits and how to correctly calculate them is an essential skill in calculus, particularly when examining the continuity of functions at a specific point.