Question

In Exercises 15–22, use the graph to find the limit (if it exists). If the limit does not exist, explain why. $$ \lim _{x \rightarrow 2} \frac{|x-2|}{x-2} $$

Short answer

The limit of the given function as \(x\) approaches 2 does not exist because the limit from the left side is not equal to the limit from the right side.

Step-by-step solution

Calculate the limit as x approaches from the left

When \(x\) approaches 2 from the left (i.e. \(x<2\)), \(|x-2|\) is negative, thus the whole equation turns into \(-1\). So, we have \(\lim_{{x \rightarrow 2}^{-}} \frac{|x-2|}{x-2} = -1\).

Calculate the limit as x approaches from the right

When \(x\) approaches 2 from the right (i.e. \(x>2\)), \(|x-2|\) is positive, thus the whole equation turns into \(1\). So, we have \(\lim_{{x \rightarrow 2}^{+}} \frac{|x-2|}{x-2} = 1\).

Comparing the left-hand limit and right-hand limit

The limit of a function at a certain point exists only if the left-hand limit and the right-hand limit at that point are equal. Here, \(\lim_{{x \rightarrow 2}^{-}} \frac{|x-2|}{x-2} \neq \lim_{{x \rightarrow 2}^{+}} \frac{|x-2|}{x-2}\). Therefore, the limit at \(x=2\) does not exist.

Key concepts

Limits

The concept of limits is central to calculus and is used to understand the behavior of functions as they approach a particular point. A limit examines what value a function seems to be getting closer to as the input approaches a specific value. For example, when we say find the limit of a function as \( x \) approaches \( 2 \), we're determining what value the function approaches when \( x \) is nearly \( 2 \). Limits help us handle indeterminate forms and situations where functions do not have an easily obtainable output by plugging in a number directly.

Mathematically, a limit can be expressed by \( \lim_{{x \to a}} f(x) = L \), meaning as \( x \) gets closer and closer to \( a \), the function \( f(x) \) gets closer and closer to \( L \). However, for a limit to exist, both the one-sided limits (from the left and from the right) at that point must be equal.

One-sided limits

One-sided limits are a way to evaluate the limit of a function as it approaches a specific point from one side: the left or right. This is crucial for functions that behave differently when approached from different directions.

• The left-hand limit examines the behavior as \( x \) approaches a point from values less than the point. It is denoted by \( \lim_{{x \to a^-}} f(x) \).
• The right-hand limit checks the function's behavior as \( x \) approaches from larger values, represented by \( \lim_{{x \to a^+}} f(x) \).

For the limit of a function at a point \( a \) to exist, both the left-hand and right-hand limits must not only exist but also be equal. If they differ, the overall limit at that point does not exist. By examining one-sided limits in our example, we observed that the left-hand limit was \(-1\) while the right-hand limit was \(1\), proving the limit did not exist.

Piecewise functions

Piecewise functions are defined by different expressions for different intervals of the domain. They generally have distinct rules depending on the input value, making them very versatile in modeling scenarios where behavior changes over time or conditions.

For example, a piecewise function might behave one way for inputs \( x<2 \) and another for \( x>2 \). These differences can lead to interesting limit behavior, especially at the points of transition. When evaluating limits, especially one-sided limits at transition points, distinct expressions of a piecewise function must be carefully examined. This means for each side of a point, different expressions in the function could result in different output behaviors. Hence, the importance of understanding one-sided limits when dealing with piecewise functions.

Absolute value

The absolute value of a number signifies its non-negative value or distance from zero. In mathematical notation, the absolute value of \( x \), represented as \( |x| \), is:

• Equal to \( x \) if \( x \) is positive or zero.
• Equal to \( -x \) if \( x \) is negative.

This causes a piecewise behavior in functions involving absolute values because the expression changes depending on whether the input is positive or negative.

In the case of our example, \( |x-2| \) results in different behaviors when \( x<2 \) and \( x>2 \). This shift, due to the absolute value, leads to the different one-sided limit results, highlighting the step-change characteristic of expressions involving absolute values. Understanding how absolute value can affect function behavior is key in calculus, often leading to important insights about function limits and continuity.