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Chapter 1: Problem 19

Use numerical and graphical evidence to conjecture whether the limit at \(x=a\) exists. If not, describe what is happening at \(x=a\) graphically. $$\lim _{x \rightarrow 0} \sin \left(\frac{1}{x}\right)$$

Short Answer

Expert verified
The limit of the function \(\sin\left(\frac{1}{x}\right)\) as \(x\) approaches \(0\) does not exist. This conclusion is drawn from the fact that the function oscillates rapidly between -1 and 1 as \(x\) gets closer to \(0\), as noted both numerically and graphically.

Step by step solution

01

Understand the Problem

We are given the function \(\sin\left(\frac{1}{x}\right)\) and we need to determine if the limit exists as \(x\) approaches \(0\). We understand that the sine function oscillates between -1 and 1 for any input value. However, when the input is \(\frac{1}{x}\), as \(x\) approaches \(0\), this input becomes very large, leading to the sine function oscillating rapidly.
02

Numerical Approach

Let us approach it numerically by substituting different values closer to \(0\) (on both sides of \(0\)) in the function. We'll notice that as \(x\) gets arbitrarily close to \(0\), the function \(\sin\left(\frac{1}{x}\right)\) oscillates wildly and doesn't approach any particular value. Thus, it suggests that the limit as \(x\) approaches \(0\) does not exist.
03

Graphical Approach

Graph the function \(\sin\left(\frac{1}{x}\right)\). Observe the behavior as the graph gets closer to \(x=0\). The graph will show the function fluctuating rapidly between -1 and 1 as \(x\) approaches \(0\), reinforcing our earlier analysis. Thus, confirming the limit as \(x\) approaches \(0\) for \(\sin\left(\frac{1}{x}\right)\) does not exist.

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Key Concepts

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

Oscillation of Functions
When we talk about the oscillation of functions in calculus, we're referring to the behavior of functions that repeatedly move above and below a certain value. A classic example is the sine function, which oscillates between -1 and 1 for all its input values. This up-and-down behavior is a form of wave pattern, common to trigonometric functions like sine and cosine.

When you see a function written as \(\sin\left(\frac{1}{x}\right)\), the oscillation becomes particularly interesting as \(x\) approaches 0. Here's why: as \(x\) gets smaller, \(\frac{1}{x}\) gets larger, causing the sine function to oscillate more rapidly without settling on a single value. This intense fluctuation is visually apparent in a graph of the function and numerically evident when attempting to find the limit at \(x=0\).
Numerical Approximation
Numerical approximation is a method used in calculus to estimate the values of functions at points where an analytical calculation might be complex or impossible. It involves substituting values close to the point of interest into the function and observing the outcome. This technique is highly useful when you're dealing with limits, particularly when a function behaves erratically around that point.

In our exercise example, substituting numbers closer and closer to 0 on either side into \(\sin\left(\frac{1}{x}\right)\) produces results that bounce between -1 and 1 erratically. This numeric method gives us a snapshot of the function's behavior near \(x=0\), suggesting the absence of a limit due to the lack of a consistent approach to any single value.
Graphical Analysis
Graphical analysis involves examining the visual representation of a function to understand its behavior, particularly the limits. A graph provides an intuitive picture of what is happening as a variable approaches a particular value. By plotting \(\sin\left(\frac{1}{x}\right)\) on a graph, we see firsthand how this function vibrates with increasing intensity as it gets closer to the y-axis (where \(x=0\)).

These fluctuations grow so intense that the function appears to violently shake back and forth. This behavior - visible on a graph - confirms what numerical approximation and an understanding of oscillating functions suggest: that the limit of \(\sin\left(\frac{1}{x}\right)\) as \(x\) approaches 0 does not exist. The graphical approach is invaluable for illustrating the complex behavior of functions like these, which are challenging to describe numerically or analytically.

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Most popular questions from this chapter

Consider the following arguments concerning \(\lim _{x \rightarrow 0^{+}} \sin \frac{\pi}{x}\) First, as \(x>0\) approaches \(0, \frac{\pi}{x}\) increases without bound; since \(\sin t\) oscillates for increasing \(t\), the limit does not exist. Second: taking \(x=1,0.1,0.01\) and so on, we compute \(\sin \pi=\sin 10 \pi=\sin 100 \pi=\cdots=0 ;\) therefore the limit equals \(0 .\) Which argument sounds better to you? Explain. Explore the limit and determine which answer is correct.

Find a function \(g(x)\) such that \(f(x)=\frac{x-4}{g(x)}\) has two horizontal asymptotes \(y=\pm 1\) and no vertical asymptotes.

Sketch the graph of \(f(x)=\left\\{\begin{array}{ll}x^{3}-1 & \text { if } \quad x<0 \\ 0 & \text { if } \quad x=0 \\ \sqrt{x+1}-2 & \text { if } \quad x>0\end{array}\right.\) and identify each limit. (a) \(\lim _{x \rightarrow 0^{-}} f(x)\) (b) \(\lim _{x \rightarrow 0^{+}} f(x)\) (c) \(\lim _{x \rightarrow 0} f(x)\) (d) \(\lim _{x \rightarrow-1} f(x)\) (e) \(\lim _{x \rightarrow 3} f(x)\)

It is very difficult to find simple statements in calculus that are always true; this is one reason that a careful development of the theory is so important. You may have heard the simple rule: to find the vertical asymptotes of \(f(x)=\frac{g(x)}{h(x)},\) simply set the denominator equal to \(0 \text { [i.e., solve } h(x)=0] .\) Give an example where \(h(a)=0\) but there is not a vertical asymptote at \(x=a\)

Evaluate \(f(-1.5), f(-1.1), f(-1.01)\) and \(f(-1.001),\) and conjecture a value for \(\lim _{x \rightarrow-1^{-}} f(x)\) for \(f(x)=\frac{x+1}{x^{2}-1} .\) Evaluate \(f(-0.5), f(-0.9), f(-0.99)\) and \(f(-0.999),\) and conjecture a value for \(\lim _{x \rightarrow-1^{+}} f(x)\) for \(f(x)=\frac{x+1}{x^{2}-1} .\) Does \(\lim _{x \rightarrow-1} f(x)\) exist?
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