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Chapter 13: Problem 46

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value. \(\lim _{x \rightarrow 1} f(x),\) where \(f(x)=\left\\{\begin{array}{ll}3 x-5 & \text { if } x \leq 1 \\ 6-2 x & \text { if } x>1\end{array}\right.\)

Short Answer

Expert verified
The limit does not exist.

Step by step solution

01

Understanding the problem

The function \( f(x) \) is piecewise, with \( f(x) = 3x - 5 \) when \( x \leq 1 \) and \( f(x) = 6 - 2x \) when \( x > 1 \). We need to determine if the limit of \( f(x) \) as \( x \) approaches 1 exists, and if it does, find its value.
02

Analyzing from the left of 1

Evaluate \( f(x) \) as \( x \) approaches 1 from the left (\( x \to 1^- \)). Since \( f(x) = 3x - 5 \) when \( x \leq 1 \), substitute \( x = 1 \) to find \( f(1) = 3(1) - 5 = -2 \). This is the value of the function as you approach from the left.
03

Analyzing from the right of 1

Evaluate \( f(x) \) as \( x \) approaches 1 from the right (\( x \to 1^+ \)). Here, \( f(x) = 6 - 2x \) since \( x > 1 \). Substitute \( x = 1 \) to get \( 6 - 2(1) = 4 \). This is the value of the function as \( x \) approaches 1 from the right.
04

Compare the left and right limits

For the limit to exist, the left-hand limit and right-hand limit must be equal. We found the left limit to be \(-2\) and the right limit to be \(4\). Since \(-2 eq 4\), the limits do not match.
05

Conclusion

Since the left-hand limit and the right-hand limit are not equal, the limit \( \lim _{x \rightarrow 1} f(x) \) does not exist.

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Key Concepts

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

Piecewise Functions
Piecewise functions are mathematical expressions defined by multiple sub-functions, each of which applies to a specific interval of the input domain. This allows us to describe a function that behaves differently in different parts of its input range. For instance, in the function given in the problem,
  • When the input, or \( x \), is less than or equal to 1, the function is defined as \( 3x - 5 \).
  • When \( x \) is greater than 1, the function changes to \( 6 - 2x \).
Understanding how a piecewise function works is crucial for analyzing limits and discontinuities, as the behavior of the function can change dramatically at the boundaries defined by the conditions of the sub-functions.
Left-Hand Limit
The left-hand limit is the value that a function approaches as the input approaches a specified point from the left side (i.e., from values less than the point of interest). In the given problem, we are interested in the behavior as \( x \) approaches 1 from values less than 1, denoted as \( x \to 1^- \).
For the piecewise function provided, when \( x \leq 1 \), the function is defined as \( 3x - 5 \). Therefore, to find the left-hand limit, we substitute \( x = 1 \) into the expression \( 3x - 5 \). This gives us:
  • \( 3(1) - 5 = -2 \)
This result, \(-2\), is the value that the function approaches from the left as \( x \) nears 1.
Right-Hand Limit
The right-hand limit considers the value that a function approaches as the input comes from values greater than a specified point. For this exercise, we need to evaluate the limit as \( x \to 1^+ \).
In the piecewise function, the relevant segment for \( x > 1 \) is \( 6 - 2x \). To compute the right-hand limit, we substitute \( x = 1 \) into the expression \( 6 - 2x \). This results in:
  • \( 6 - 2(1) = 4 \)
Thus, for values of \( x \) approaching 1 from the right, the function approaches \( 4 \). This indicates how the function behaves as it nears the point from the right side.
Discontinuity Analysis
Discontinuity analysis involves examining whether a function has any jumps or breaks at a particular point. A key aspect of this analysis is comparing left-hand and right-hand limits to determine if they are the same.
In the exercise at hand, we found:
  • The left-hand limit as \( x \to 1^- \) is \(-2\).
  • The right-hand limit as \( x \to 1^+ \) is \( 4 \).
Since \(-2 eq 4\), the limits from the left and right do not coincide at \( x = 1 \). As a result, the limit \( \lim_{x \to 1} f(x) \) does not exist, indicating a discontinuity at \( x = 1 \). This discontinuity is characteristic of piecewise functions where the sub-functions do not align seamlessly at the boundary.

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Most popular questions from this chapter

Solve each problem. W(t)$ represents the gallons of water in a tank after I minutes. Complete the following. (a) Find the initial amount of water in the tank. (b) Find the amount of water in the rank after 12 minutes. (c) Is the rate of change in the amount of water in the rank constant? Explain. (d) Find the rate of change in the amount of water at 12 minutes. $$W(t)=500-t^{2}, \quad \text { for } 0 \leq t \leq 20$$

Approximate the area under the graph of \(f(x)\) and above the \(x\) -axis, using each of the following methods with \(n=4\). (a) Use left endpoints. (b) Use right endpoints. (c) Average the answers in parts ( \(a\) ) and ( \(b\) ). (d) Use midpoints. $$f(x)=\frac{2}{x} \text { from } x=1 \text { to } x=9$$

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value. \(\lim _{x \rightarrow 0} \tan \frac{1}{x}\)

For the given \(f(x)\), find a formula for \(f^{\prime}(a)\). $$f(x)=3 x-1$$

Solve each problem. The weight of a small fish in grams after \(t\) weeks is modeled by $$W(t)=0.1 t^{2}$$. At what rate is the fish growing at time \(t=4 ?\)
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