Problem 55 - Precalculus with Limits

Question

In Exercises 55-62, graph the function. Determine the limit (if it exists) by evaluating the corresponding one-sided limits. $$\lim_{x \to 6} \dfrac{|x-6|}{x-6}$$

Step-by-Step Solution

Answer

The limit of the function as \(x\) approaches 6 does not exist.

Step 1: Identify the Function

The function presented is \(f(x) = \dfrac{|x-6|}{x-6}\). It involves an absolute value and fraction, which will influence its behavior, particularly around \(x = 6\).

Step 2: Understand the Function Behavior

The absolute value, \(|x-6|\), returns the magnitude (or 'positive value') of \(x-6\).\nWhen \(x<6\), \(x-6\) is negative, so \(|x-6|\) would be \(-(x-6)\), making the function \(f(x)= -1\).\nWhen \(x>6\), \(x-6\) is positive, so \(|x-6|\) would be \(x-6\), making the function \(f(x) = 1\).

Step 3: Graph the Function

The function breaking point is at \(x=6\) which is not included in the function domain. When \(x<6\), graph a horizontal line at \(y=-1\), and when \(x>6\), graph a horizontal line at \(y=1\).

Step 4: Determine the one-sided limits

The one-sided limit as \(x\) approaches 6 from the left (\(x<6\)) is \(\lim_{x \to 6^-} f(x) = -1\).\nThe one-sided limit as \(x\) approaches 6 from the right (\(x>6\)) is \(\lim_{x \to 6^+} f(x) = 1\).

Step 5: Determine the limit

Since the one-sided limits are not equal, the limit does not exist. Thus, \(\lim_{x \to 6} \dfrac{|x-6|}{x-6}\) does not exist.

Key Concepts

Absolute Value Function

An absolute value function takes any number and makes it non-negative. It's written \(|x|\), where \(x\) is a number or expression. This function measures the distance from zero on a number line. For example, \(|-3| = 3\) because the distance between -3 and 0 is 3.

With the function \(f(x) = \dfrac{|x-6|}{x-6}\), the absolute value is applied to \(x - 6\). This means that whether \(x - 6\) is positive or negative, \(\left|x - 6\right|\) will be positive.

• For \(x < 6\), \(x - 6\) is negative, and \(\left|x - 6\right| = -(x - 6)\).
• For \(x > 6\), \(x - 6\) is positive, so \(\left|x - 6\right| = x - 6\).

Understanding this behavior is crucial when working with absolute value functions.

One-Sided Limits

One-sided limits involve looking at the behavior of a function as it approaches a point from one side only. These are essential when a function behaves differently on each side of a certain point.

For instance, the one-sided limits of \(\dfrac{|x-6|}{x-6}\) as \(x\) approaches 6 show different behaviors for \(x < 6\) and \(x > 6\):

• **Left-hand limit:** For \(x < 6\), \(f(x) = -1\), so \(\lim_{x \to 6^-} f(x) = -1\).
• **Right-hand limit:** For \(x > 6\), \(f(x) = 1\), so \(\lim_{x \to 6^+} f(x) = 1\).

One-sided limits are powerful for understanding the specific behavior of functions at points of interest, such as discontinuities.

Limit Does Not Exist

A limit might not exist when a function does not approach a specific value as \(x\) approaches a point. This commonly occurs when the left-hand limit and right-hand limit at a point are not equal.

Consider the function \(f(x) = \dfrac{|x-6|}{x-6}\):

• The left-hand limit at \(x = 6\) is \(-1\).
• The right-hand limit at \(x = 6\) is \(+1\).

Because these one-sided limits differ, \(\lim_{x \to 6} \dfrac{|x-6|}{x-6}\) does not exist. The concept of a limit not existing is key in calculus, especially with piecewise functions and absolute values.

Graphing Piecewise Functions

Graphing piecewise functions helps visualize how a function behaves over different intervals. Each piece of the function is graphed separately.

For the function \(f(x) = \dfrac{|x-6|}{x-6}\), the graph has two horizontal lines:

• For \(x < 6\), the graph is a line at \(y = -1\).
• For \(x > 6\), the graph is a line at \(y = 1\).

At \(x = 6\), there's a break in the graph since \(\dfrac{|x-6|}{x-6}\) is undefined there.
Using different pieces simplifies visualizing complex functions, revealing discontinuities and limits.