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Chapter 2: Problem 43

Give the four limil statcments displayed in (2.2.6), taking f(x)=\frac{1}{x-1}, c=3

Short Answer

Expert verified
The four limit statements for the function \(f(x)=\frac{1}{x-1}\) at \(x = 3\) are: \( \lim_{{x \rightarrow 3^-}} f(x) = \frac{1}{2}, \lim_{{x \rightarrow 3^+}} f(x) = \frac{1}{2}, \lim_{{x \rightarrow 3}} f(x) = \frac{1}{2}, \) and \( f(3) = \frac{1}{2} \).

Step by step solution

01

Calculating the limit as x approaches the value c from the left

We will be calculating the limit of \( f(x) \) as \( x \) approaches \( c \) from the left side (\( x \rightarrow 3^-\)). To do this, substitute \( x = 3 \) into the function \( f(x) \), so: \( \lim_{{x \rightarrow 3^-}} f(x) = \frac{1}{{(3 - 1)}} \). Which simplifies to \( \frac{1}{2} \).
02

Calculating the limit as x approaches the value c from the right

Next, we'll determine the limit of \( f(x) \) as \( x \) approaches \( c \) from the right side (\( x \rightarrow 3^+ \)). For this, substitute \( x = 3 \) into the function \( f(x) \), resulting in: \( \lim_{{x \rightarrow 3^+}} f(x) = \frac{1}{{(3 - 1)}} \). The simplification of this limit is also \( \frac{1}{2} \).
03

Calculating the overall limit

The overall limit of \( f(x) \) as \( x \) approaches \( c \) from both the left and the right is now calculated. Because the left and right limits match at \( \frac{1}{2} \), we can say that the overall limit as \( x \) approaches \( c \) from both sides is \( \frac{1}{2} \). Thus: \( \lim_{{x \rightarrow 3}} f(x) = \frac{1}{2} \).
04

Find the function value at x = c

Finally, calculate the function value \( f(c) \). Substitute \( x = 3 \) into the function \( f(x) \), which is \( f(3) = \frac{1}{{(3 - 1)}} \). The value for this is \( \frac{1}{2} \).

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Key Concepts

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

One-sided limits
When exploring calculus, understanding one-sided limits is fundamental. They help us get a clearer view of the behavior of a function as it approaches a certain point from one specific direction. Imagine approaching a door and deciding to either open it from the left or the right; similarly, one-sided limits analyze a function as the x-values approach a specified point from only one side.

There are two varieties of one-sided limits:
  • Left-hand limit (\( x \rightarrow c^- \)): This looks at the values that the function approaches as x gets closer to a particular number, c, from the left or smaller values. In our exercise, as x approaches 3 from the left, we substitute directly into the function: \( \lim_{{x \rightarrow 3^-}} f(x) = \frac{1}{{(3 - 1)}} \), which simplifies to \( \frac{1}{2} \).
  • Right-hand limit (\( x \rightarrow c^+ \)): This evaluates the function as x approaches the number from the right or larger values. Using our example, we check: \( \lim_{{x \rightarrow 3^+}} f(x) = \frac{1}{2} \), confirming the function behaves consistently from the right.
In essence, the one-sided limits tell us the direction-specific behavior of \( f(x) \) as it nears a specified point like 3 in our problem.
Two-sided limits
Two-sided limits blend the information received from both the left-hand and right-hand limits to find a comprehensive understanding of a function at a given input. It is a bit like observing a street from both ends to get a full picture.

To confirm a two-sided limit exists, the values from both the left and right sides as 'x' approaches must align. This indicates the function is behaving uniformly from both directions.
  • Mathematically, this is expressed as \( \lim_{{x \rightarrow c}} f(x) \). If \( \lim_{{x \rightarrow c^-}} f(x) = \lim_{{x \rightarrow c^+}} f(x) \), then a two-sided limit is present.
  • Our problem confirms this reality, with both sides approaching \( \frac{1}{2} \). Thus, \( \lim_{{x \rightarrow 3}} \frac{1}{x-1} = \frac{1}{2} \) represents the entire limit as 'x' hones in on 3.
Two-sided limits are crucial because they often indicate the presence of continuity, a key characteristic in calculus, portraying a seamless shift through a particular x-value.
Function evaluation
Function evaluation is the process of determining the output of a function for a specific input. It鈥檚 akin to feeding data into a machine to see what result is produced.

In our scenario, we are tasked with calculating \( f(x) \) at a specific point to get \( f(c) \). Particularly, this involves substituting \( x = 3 \) into the formula \( f(x) = \frac{1}{x-1} \).
  • By performing direct substitution, you determine \( f(3) = \frac{1}{3-1} = \frac{1}{2} \).
  • Unlike limits, function evaluation does not describe approaching behavior but gives an exact point value.
Understanding function evaluation is vital as it helps in comprehending exact values, which is crucial for solving various real-world problems and calculus exercises.

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Most popular questions from this chapter

Give necessary and sufficient conditions on \(A\) and \(B\) for the function $$f(x)=\left\\{\begin{aligned} A x-B, & x \leq 1 \\ 3 x, & 1 < x \cdot 2 \\ B x^{2}-A, & 2 \leq x \end{aligned}\right.$$ to be continuous at \(x=1\) but discontinuous at \(x=2\).

Evaluate \(\lim _{h \rightarrow 0}[f(c+h)-f(c)] / h\). \(f(x)=\sin x, c=\pi / 4,\) HINT: Use the addition formula for the sine function.

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If possible, define the function at 1 so that it becomes continuous at 1. $$f(x)=\frac{x^{2}-1}{x-1}$$.

Let \(f\) be a function defined on \(\operatorname{some}\) interval \((c-p, c+p) .\) Now change the value of \(f\) at a finite number of points \(x_{1}, x_{2}, \ldots ., x_{n}\) and call the resulting function \(g\) $$\text { (a) Stow that if } \lim _{x \rightarrow c} f(x)=L, \text { then } \lim _{x \rightarrow c} g(x)=L$$ (b) Show that if lim does not exist, then \(\lim _{x \rightarrow c} g(x)\) does not 驴xist.
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