Question

Absolute value Show that \(\lim _{x \rightarrow 0}|x|=0\) by first evaluating \(\lim _{x \rightarrow 0^{-}}|x|\) and \(\lim _{x \rightarrow \infty}|x| .\) 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 limit as x approaches 0 from the left (0-)

To find \(\lim_{x\to 0^{-}}|x|\), we need to substitute x with values that are slightly less than 0 (e.g., -0.1, -0.01, -0.001, etc.). According to the definition of the absolute value function, if x is less than 0, then \(|x| = -x\). So, for x approaching 0 from the left: $$\lim_{x\to 0^{-}}|x| = \lim_{x\to 0^{-}}(-x).$$ As x approaches to 0 from the left, -x becomes closer to 0. Thus, the limit is: $$\lim_{x\to 0^{-}}(-x) = 0.$$

Evaluate limit as x approaches 0 from the right (0+)

To find \(\lim_{x\to 0^{+}}|x|\), we need to substitute x with values that are slightly greater than 0 (e.g., 0.1, 0.01, 0.001, etc.). According to the definition of the absolute value function, if x is greater than or equal to 0, then \(|x| = x\). So, for x approaching 0 from the right: $$\lim_{x\to 0^{+}}|x| = \lim_{x\to 0^{+}}(x).$$ As x approaches to 0 from the right, x becomes closer to 0. Thus, the limit is: $$\lim_{x\to 0^{+}}(x) = 0.$$

Evaluate the limit as x approaches 0

Now that we have the limit of the absolute value function as x approaches 0 from both the left and the right, we can use these values to evaluate the limit as x approaches 0. Since \(\lim_{x\to 0^{-}}|x| = 0\) and \(\lim_{x\to 0^{+}}|x| = 0\), it follows that the limit of the absolute value function as x approaches 0 is: $$\lim_{x\to 0}|x| = 0$$ We have proved that \(\lim_{x\to 0}|x|=0\).

Key concepts

Calculus

Calculus is an area of mathematics that studies how things change. It provides a framework for modeling systems in which there is change, and a way to deduce the predictions of such models. One of its main operations is the limit, which allows us to study the behavior of functions as they approach a certain point or value.

Within calculus, limits help us understand the behavior of functions at points where they are not necessarily defined or where they exhibit unusual behavior, such as near sharp turns or asymptotes. This concept is fundamental as it forms the foundation of other crucial topics in calculus, such as derivatives and integrals.

Limits

Limits are a core concept in calculus that describes the value that a function approaches as the input approaches some value. Limits can be approached from both the left and the right, and these are known as one-sided limits. When both one-sided limits are the same, the two-sided limit exists and is equal to that common value.

To say that \(\lim_{x\to a}f(x) = L\) means that as x gets closer and closer to 'a', the function f(x) gets closer and closer to the value L. Limits can be used to describe the behavior of functions infinitely close to a particular point, even if they're not explicitly defined at that point.

Absolute Value

The absolute value function is written as \( |x| \) and is defined as the distance from zero on the real number line without regard to direction. For all real numbers x, the absolute value function is defined by:
\[|x| = \left\{\begin{array}{ll}x & \text{if } x \geq 0 \-x & \text{if } x<0\end{array}\right.\]
The function \( |x| \) is always nonnegative and reflects the magnitude of a number, regardless of its sign. When dealing with limits involving the absolute value function, it's crucial to consider the behavior from both sides of the point of interest.

One-Sided Limits

One-sided limits refer to the value that a function approaches as the input approaches a specified point from one side: either from the left (denoted by \(x\to c^- \) ) or from the right \(x\to c^+ \) ). Such limits are crucial when analyzing the behavior of functions that are not symmetric around a point. The need to consider one-sided limits arises especially when dealing with functions that have absolute values because the definition of the absolute value function changes based on whether the input is positive or negative.

In the case of the absolute value function, as seen in the provided exercise, evaluating one-sided limits becomes essential to determining whether the limit exists at a point and what that limit is.