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Magazine Chapter 1: Problem 81
Let \(f(x)=(1 / x) \sin (1 / x) .\) Is the statement true or false? Explain. The function \(f(x)\) has a vertical asymptote at \(x=0\).
Short Answer
Expert verified
True, the function has a vertical asymptote at \(x=0\).
Step by step solution
01
Understanding Vertical Asymptotes
A function has a vertical asymptote at a point if, as you approach that point, the function's values grow infinitely large or infinitely small. This typically happens at points where the function is undefined.
02
Examine the Function at x=0
The function is given as \[ f(x) = \frac{1}{x} \sin \left(\frac{1}{x}\right) \]Note that it is undefined at \(x=0\) because it involves division by zero, which suggests a potential vertical asymptote.
03
Behavior of f(x) as x Approaches 0
As \(x\) approaches 0, both \(\frac{1}{x}\) and \(\sin \left(\frac{1}{x}\right)\) contribute to the behavior of \(f(x)\). The \(\frac{1}{x}\) term suggests very large values when \(x\) is near 0, causing oscillations in \(\sin \left(\frac{1}{x}\right)\) to be magnified.
04
Analyze Using Limits
Consider the limit:\[\lim_{x \to 0} \frac{1}{x} \sin\left(\frac{1}{x}\right)\]Since \(\sin\left(\frac{1}{x}\right)\) oscillates between -1 and 1, the product \(\frac{1}{x} \sin\left(\frac{1}{x}\right)\) does not approach any finite limit or reduce to 0. Instead, it grows without bound, indicating a vertical asymptote.
05
Conclusion
Since \(f(x)\) becomes infinitely large or small as \(x\) approaches 0 without settling on a finite limit, the function does have a vertical asymptote at \(x=0\). Thus, the statement is true.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Limits and Continuity
When learning about limits and continuity, it鈥檚 important to understand how a function behaves as it approaches a certain point. In simple terms, a limit tells us how a function behaves when getting really close to a specific value on the X-axis.
For instance, consider the function \[ f(x) = \frac{1}{x} \sin \left(\frac{1}{x}\right) \]at values close to zero. The limit of this function helps us determine if there is a continuity or a break, such as an infinite jump, near that point.
For instance, consider the function \[ f(x) = \frac{1}{x} \sin \left(\frac{1}{x}\right) \]at values close to zero. The limit of this function helps us determine if there is a continuity or a break, such as an infinite jump, near that point.
- To assess this, we particularly focus on what happens as values of \(x\) approach zero from both the positive and negative sides.
- If the values of \(f(x)\) tend to stabilize around a single number, we say the function has a finite limit and is continuous there.
- However, if the values grow wildly large or endlessly small, then the function has either a vertical asymptote or some discontinuity at that point.
Asymptotic Behavior
Asymptotic behavior is like a mathematical detective story, revealing how a function behaves as the variable approaches certain critical points such as infinity or zero.
Understanding this helps in identifying points such as vertical or horizontal asymptotes, where the curve of the graph goes towards but never quite touches a specific line or rises to infinity.
Understanding this helps in identifying points such as vertical or horizontal asymptotes, where the curve of the graph goes towards but never quite touches a specific line or rises to infinity.
- Vertical asymptotes occur at certain \(x\)-values where the function grows infinitely either positively or negatively. This tells us that the function is undefined at that point, like in the given function at \(x=0\).
- Essentially, you look at how \( f(x) = \frac{1}{x} \sin \left(\frac{1}{x} \right) \) behaves, examining the multiplicative relationships.
- The term \(\frac{1}{x}\) blows up to very large values as \(x\) nears zero, causing the overall function to behave erratically, confirming the vertical asymptotic behavior.
Trigonometric Functions
Trigonometric functions, like sine and cosine, are fundamental in understanding oscillations and waves in mathematics. They operate under periodic behavior, looping their values within a fixed range, typically between -1 and 1 for sine and cosine.
This property affects how they behave, especially when intertwined with other functions. For instance, in our function:
This property affects how they behave, especially when intertwined with other functions. For instance, in our function:
- \(\sin(\frac{1}{x})\) rapidly oscillates between -1 and 1 as \(x\) approaches zero.
- These frequent oscillations magnify due to the multiplication by \(\frac{1}{x}\), leading to larger swings in value.
- This is a significant factor in the formation of the vertical asymptote, since these oscillatories don't allow \(f(x)\) to stabilize around a particular value.
