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Chapter 12: Problem 55

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

Short Answer

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

Step by step solution

01

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 \(x = 6\).
02

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 \(f(x) = 1\).
03

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 \(y=1\).
04

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 \(\lim_{x \to 6^+} f(x) = 1\).
05

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

In Exercises 9-28, find the limit (if it exists). If the limit does not exist, explain why. Use a graphing utility to verify your result graphically. \\[\lim_{x\to \infty} \dfrac{1-2x}{x+2} \\]

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