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Chapter 1: Problem 36

Use algebra to evaluate \(\lim _{x \rightarrow 0} f(x), \lim _{x \rightarrow a} f(x),\) and \(\lim _{x \rightarrow a} f(x)\) if they exist. Sketch a graph to confirm your answers. $$a=4, \quad f(x)=\frac{|x-4|}{x-4}$$

Short Answer

Expert verified
\(\lim_{x \to 0} f(x) = -1\), \(\lim_{x \to 4} f(x)\) does not exist.

Step by step solution

01

Understanding the Function

The function given is \(f(x) = \frac{|x-4|}{x-4}\). This function is defined for all \(x eq 4\), since at \(x = 4\), the denominator becomes zero, making the function undefined.
02

Evaluating the One-Sided Limits at x = 4

We need to consider the behavior of \(f(x)\) as \(x\) approaches 4 from both left (\(x \to 4^-\)) and right (\(x \to 4^+\)) separately. 1. **Left Limit:** For \(x < 4\), \(|x-4|=-(x-4)\), so the function becomes \(f(x) = \frac{-(x-4)}{x-4} = -1\). 2. **Right Limit:** For \(x > 4\), \(|x-4|=(x-4)\), thus \(f(x) = \frac{x-4}{x-4} = 1\).
03

Determining Lim \(x \to a\) where \(a = 4\)

Since the left-hand limit as \(x\) approaches 4 is \(-1\) and the right-hand limit is \(1\), the two limits do not equal each other. Therefore, \(\lim_{x \to 4}f(x)\) does not exist.
04

Evaluating Lim \(x \to 0\)

Since \(f(x) = \frac{|x-4|}{x-4}\), the value of \(x\) approaching 0 does not affect the independent limits evaluated at 4. Because the function is only undefined at \(x = 4\) and defined elsewhere, \(\lim_{x \to 0} f(x) = \frac{|0-4|}{0-4} = \frac{4}{-4} = -1\).
05

Graphing the Function

The graph of \(f(x) = \frac{|x-4|}{x-4}\) can be visualized as having two pieces: - For \(x < 4\), it is a horizontal line \(y = -1\).- For \(x > 4\), it is a horizontal line \(y = 1\).- At \(x = 4\), the function is undefined, creating a gap or hole.

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Key Concepts

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

One-sided limits
Understanding one-sided limits is crucial when examining how a function behaves as it approaches a certain point from either the left or the right. For the function given, \(f(x) = \frac{|x-4|}{x-4}\), we analyzed its behavior as \(x\) tends toward 4. There are two one-sided limits to consider here:
  • **Left-hand Limit:** This is evaluated when \(x \rightarrow 4^-\). For \(x < 4\), we have \(|x-4| = -(x-4)\), simplifying the function to \(-1\). Thus, \(\lim_{x \to 4^-} f(x) = -1\).
  • **Right-hand Limit:** This is evaluated when \(x \rightarrow 4^+\). For \(x > 4\), \(|x-4| = x-4\), so the function simplifies to \(1\). Hence, \(\lim_{x \to 4^+} f(x) = 1\).
The distinction between these values highlights that the left and right limits are not equal, a factor leading to discontinuity.
Piecewise function
A piecewise function is defined by different expressions for different intervals of the domain. In our case, the function \(f(x) = \frac{|x-4|}{x-4}\) behaves differently depending on whether \(x\) is less than or greater than 4. This results in two separate constant functions:
  • For \(x < 4\), the function piece is: \(f(x) = -1\).
  • For \(x > 4\), the function piece is: \(f(x) = 1\).
Piecewise functions like this are very useful in mathematics to describe situations where a function exhibits different behavior over different intervals. Each 'piece' acts like its own mini-function defining behavior over a specific range.
Function graphing
Graphing the function \(f(x) = \frac{|x-4|}{x-4}\) can aid in visualizing how different pieces of a piecewise function manifest. When graphing this function:
  • For \(x < 4\), draw a horizontal line at \(y = -1\).
  • For \(x > 4\), draw a horizontal line at \(y = 1\).
By graphing these lines, you visualize how the function exists on different levels of the y-axis beyond the point \(x = 4\). Remember, the function is undefined at \(x = 4\), represented by a hole or gap on the graph right at the vertical line \(x = 4\). This graph effectively communicates how the function shifts between values as \(x\) changes, offering a visual understanding of both piecewise definition and the discontinuity.
Discontinuity in functions
A discontinuity in a function occurs when there is a break, gap, or hole in a graph where the function is not defined or properly continuous. In this case, \(f(x) = \frac{|x-4|}{x-4}\) exhibits a discontinuity at \(x = 4\):
  • The function is undefined at \(x = 4\) due to division by zero in the denominator.
  • The left-hand and right-hand limits at this point are not equal (\(-1\) and \(1\) respectively), leading to what is called a jump discontinuity.
Such discontinuities are important in understanding the overall behavior of functions鈥攖hey mark points where functions may have abrupt changes or be entirely undefined. Recognizing and interpreting discontinuities is a key aspect of analyzing piecewise and complex functions.

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Most popular questions from this chapter

Give an example of: A rational function that has a limit at \(x=1\) but is not continuous at \(x=1\).

For the given constant \(c\) and function \(f(x),\) find a function \(g(x)\) that has a hole in its graph at \(x=c\) but \(f(x)=g(x)\) everywhere else that \(f(x)\) is defined. Give the coordinates of the hole. $$f(x)=\ln x, c=1$$

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