Reciprocal Function is Unbounded

In summary, if the function ##f## is defined as ##f(x) = 1## for rational numbers in the interval (0,∞) and ##f(x) = x-3## for irrational numbers in the same interval, then ##\frac{1}{f}## is unbounded on every interval containing ##x=3##. However, if ##f## is required to be continuous, the theorem does not hold and ##\lim_{x\to 3} f(x) = 0## does not necessarily hold.
  • #1
Bashyboy
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Homework Statement


Let ##f : (0,\infty) \to \Bbb{R} - \{0\}##. If ##\frac{1}{f}## is unbounded on every interval containing ##x=3##, will ##\lim_{x \to 3} f(x) = 0##?

Homework Equations

The Attempt at a Solution



Consider the function

$$f(x) = \begin{cases} 1, & x \in \Bbb{Q} \cap (0,\infty) \\ x-3, & x \notin \Bbb{Q} \cap (0, \infty) \\ \end{cases}$$

Let ##x_n \notin \Bbb{Q} \cap (0,\infty)## be an irrational sequence converging to ##3##. Then ##\lim_{n \to \infty} \frac{1}{f(x_n)} = \lim_{n \to 3} \frac{1}{x_n -3} = \pm \infty## (I am abusing notation out of laziness). This shows that ##\frac{1}{f}## is unbounded on every interval of containing ##3##. Now let ##r_n \in \Bbb{Q} \cap (0, \infty)## be a rational sequence converging to ##3##. Then ##\lim_{n \to \infty} f(r_n) = \lim_{n \to \infty} 1 = 1 \neq 0##.

How does this sound? What if ##f## is required to be continuous; will the theorem hold?
 
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  • #2
Yes that is correct. If ##f## is continuous the theorem will not hold - although that needs to be proven.
 

Related to Reciprocal Function is Unbounded

What does it mean for a reciprocal function to be unbounded?

When a reciprocal function is unbounded, it means that as the input values approach certain values, the output values of the function will become infinitely large or small. This is because the function is not restricted by any upper or lower limit.

How can you tell if a reciprocal function is unbounded?

A reciprocal function is unbounded if it does not have any asymptotes or if it has a vertical asymptote at x=0. This means that the function will continue to increase or decrease without bound as x approaches 0.

What is the difference between a bounded and unbounded reciprocal function?

A bounded reciprocal function is one that is limited by an upper and lower bound. This means that the function will approach a finite value as x approaches certain values. On the other hand, an unbounded reciprocal function has no limits and the output values will continue to increase or decrease without bound as x approaches certain values.

Can a reciprocal function be both bounded and unbounded?

No, a reciprocal function cannot be both bounded and unbounded at the same time. It can only be one or the other depending on its behavior and the presence of asymptotes.

How can the unboundedness of a reciprocal function affect its graph?

The unboundedness of a reciprocal function can cause the graph to have vertical asymptotes at certain points, making it discontinuous. It can also cause the graph to approach infinity or negative infinity as x approaches certain values, resulting in a graph that appears to have a "hole" or a "jump" in it.

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