Problem 18 - Contemporary Precalculus

Question

Find the limit if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow 2^{+}} \frac{|x-2|}{x-2}$$

Step-by-Step Solution

Answer

Answer: The limit is 1.

Step 1: Understand the notation

The given limit is written as $$\lim _{x \rightarrow 2^{+}} \frac{|x-2|}{x-2}$$ Here, \(x \rightarrow 2^{+}\) means that the limit is being approached from the positive side of 2 (i.e., x is slightly greater than 2).

Step 2: Consider the absolute value

Recall that the absolute value function is defined as: $$|a| = \begin{cases} a, & \mbox{if } a \ge 0 \\ -a, & \mbox{if } a < 0 \end{cases}$$ As \(x \rightarrow 2^{+}\), we observe that \((x - 2)\) is always greater than or equal to 0 since x is slightly greater than 2. Therefore, we can rewrite the function as: $$\frac{|x-2|}{x-2} = \frac{x-2}{x-2}$$

Step 3: Simplify the expression

Notice that the numerator and denominator are equal, so we can simplify the expression as: $$\frac{x-2}{x-2} = 1$$

Step 4: Evaluate the limit

Since the expression simplifies to a constant, the limit as \(x \rightarrow 2^{+}\) is equal to that constant value. Thus, the limit is: $$\lim _{x \rightarrow 2^{+}} \frac{|x-2|}{x-2} = \boxed{1}$$

Key Concepts

Limits from the Right

In calculus, when we talk about a limit with the notation \( \lim_{x \rightarrow a^+} \), we're focusing on the behavior of a function as \( x \) approaches the number \( a \) strictly from the right-hand side. This means that \( x \) takes values slightly greater than \( a \). In our example, we're examining what happens to the expression \( \frac{|x-2|}{x-2} \) as \( x \) gets closer to 2 from the positive side. This type of limit is particularly useful for functions that behave differently from the left and right.
Understanding this is crucial because it helps us determine if functions have certain behaviors or discontinuities at points of interest.

Absolute Value Function

The absolute value function plays a key role in expressions involving modulus symbols, such as \( |x-2| \). The absolute value \( |a| \) is defined as follows:

• \( a \), if \( a \ge 0 \)
• \( -a \), if \( a < 0 \)

This means it transforms any negative input into its positive counterpart, while leaving non-negative inputs unchanged.
In the context of the limit \( \lim_{x \rightarrow 2^+} \), since \( x > 2 \), \( x-2 \) is non-negative. Consequently, \( |x-2| = x-2 \) for \( x \) just greater than 2. This detail allows us to simplify the expression significantly.

Limit Simplification

Simplifying limits often involves clever manipulation of expressions, especially when dealing with absolute values. In our example \( \frac{|x-2|}{x-2} \), simplifying becomes straightforward once we realize that \( |x-2| \) equals \( x-2 \) given \( x > 2 \).
By substituting \( x-2 \) for \( |x-2| \), the expression simplifies to \( \frac{x-2}{x-2} \).
This results in \( 1 \) because you are effectively dividing the same number by itself (provided it's non-zero). This step is critical because it turns a complex-looking limit into a simple constant that can be easily evaluated.

Greater Than Conditions

The condition \( x > 2 \) plays a vital role in this exercise. It directly affects how we interpret and simplify the absolute value expression. Since \( x \) is slightly greater than 2, \( x-2 \) is positive. Therefore, the condition \( x > 2 \) allows us to treat \( |x-2| \) simply as \( x-2 \).
Greater than conditions are essential because they help us determine which branch of a piecewise function, such as the absolute value function, should be used.
Without understanding the importance of such conditions, we might incorrectly simplify the expression, leading to wrong conclusions about the limit.