Show that there is a M>0 such that |f(x)|<M

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In summary, the conversation discusses a problem where a continuous function is given and its limits at positive and negative infinity are known. The goal is to show that there exists a value $M$ such that the function's absolute value is always smaller than $M$ for all values of $x$. The participants discuss different approaches and eventually arrive at the conclusion that the function is bounded on a closed segment, but it is not possible to find a specific value for $M$ without knowing more about the function.
  • #1
mathmari
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Hey! :eek:

I have the following exercise:
It is given that $f:R \to R$ continuous and $\lim_{x \to -\infty}f(x)=-5$ & $\lim_{x \to +\infty}f(x)=5$.Show that there is a $M>0 \in R$ such that $|f(x)|<M, \forall x \in R$.Can you find a value for $M$?

Could you tell me if that what I have tried is right or what I could change??

$$\lim_{x \to -\infty}f(x)=-5:$$
$ \forall \epsilon>0 $ there is a $c>0$ such that $\forall x<-c \Rightarrow |f(x)-(-5)|<\epsilon \Rightarrow - \epsilon < f(x)+5< \epsilon \Rightarrow - \epsilon -5< f(x)< \epsilon -5$$ $(1)
$$\lim_{x \to +\infty}f(x)=5:$$
$ \forall \epsilon ' >0$ (also for $\epsilon ' =\epsilon$) there is a $c'>0$ such that $\forall x>c' \Rightarrow |f(x)-5|< \epsilon \Rightarrow -\epsilon < f(x) -5 < \epsilon \Rightarrow 5- \epsilon < f(x)< \epsilon +5 $ $(2)$

From the relations $(1)$ and $(2)$ we get $$- \epsilon -5<f(x)< \epsilon -5 < \epsilon +5$$
$$ - \epsilon +5 <f(x)< \epsilon +5$$
So $|f(x)|< \epsilon +5$.
 
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  • #2
mathmari said:
So $|f(x)|< \epsilon +5$.
This graph (clickable) begs to differ.

[GRAPH]oatb2hhtr1[/GRAPH]
 
  • #3
Evgeny.Makarov said:
This graph (clickable) begs to differ.

[GRAPH]oatb2hhtr1[/GRAPH]

So is the way I tried to solve it wrong, or just the value of $M$?
 
  • #4
mathmari said:
\[\lim_{x \to -\infty}f(x)=-5:\]
$ \forall \epsilon>0 $ there is a $c>0$ such that $\forall x<-c \Rightarrow |f(x)-(-5)|<\epsilon \Rightarrow - \epsilon < f(x)+5< \epsilon \Rightarrow - \epsilon -5< f(x)< \epsilon -5$$ $(1)
$$\lim_{x \to +\infty}f(x)=5:$$
$ \forall \epsilon ' >0$ (also for $\epsilon ' =\epsilon$) there is a $c'>0$ such that $\forall x>c' \Rightarrow |f(x)-5|< \epsilon \Rightarrow -\epsilon < f(x) -5 < \epsilon \Rightarrow 5- \epsilon < f(x)< \epsilon +5 $ $(2)$
Up to here everything is correct. I recommend you take the graph as a counterexample and use it to find the error in the last part of your post. Pick a specific value of $\epsilon$, say, $\epsilon=1$, see if (1) and (2) are true and whether the conclusion you derive from them is true.

It is clear that the peak of the graph can be made as tall as possible, so it is not possible to give an upper bound in advance that works for all functions. Another hint is that you should use the fact that a continuous function on a closed segment is bounded (Wikipedia). The challenge is to find that closed segment.
 
  • #5
mathmari said:
It is given that $f:R \to R$ continuous and $\lim_{x \to -\infty}f(x)=-5$ & $\lim_{x \to +\infty}f(x)=5$.Show that there is a $M>0 \in R$ such that $|f(x)|<M, \forall x \in R$.Can you find a value for $M$?
You do not have to be that abstract. Use the theorem: If f is continuous on [-n,n] then it is bounded there.

From the given:
$\left( {\exists {M_1} \in {\mathbb{Z}^ + }} \right)\left[ {x \geqslant {M_1} \Rightarrow \left| {f(x) - 5} \right| < 1} \right]$ & $\left( {\exists {M_2} \in {\mathbb{Z}^ + }} \right)\left[ {x \le -{M_2} \Rightarrow \left| {f(x) + 5} \right| < 1} \right]$

In each of those cases $|f(x)|<6$.

Now let $N=M_1+M_2$. The continuous function $f$ bounded on $[-N,N]$ by some $B>0$

So $\forall x\in\mathbb{R},~|f(x)|<B+6$
 
  • #6
Evgeny.Makarov said:
Pick a specific value of $\epsilon$, say, $\epsilon=1$, see if (1) and (2) are true and whether the conclusion you derive from them is true.
For $\epsilon=1$ we get from the relation $(1)$ $-6 <f(x)<-4$, and from the relation $(2)$ we get $4<f(x)<6$.
Does this mean that we cannot say that $|f(x)|<6$?

Evgeny.Makarov said:
It is clear that the peak of the graph can be made as tall as possible, so it is not possible to give an upper bound in advance that works for all functions. Another hint is that you should use the fact that a continuous function on a closed segment is bounded (Wikipedia). The challenge is to find that closed segment.

To find that closed segment do we have to use the relations $(1)$ and $(2)$?
 
  • #7
Plato said:
Now let $N=M_1+M_2$. The continuous function $f$ bounded on $[-N,N]$ by some $B>0$

So $\forall x\in\mathbb{R},~|f(x)|<B+6$

I got stuck...Could you explain me why |f(x)|<B+6 ?
 
  • #8
mathmari said:
I got stuck...Could you explain me why |f(x)|<B+6 ?

You have three cases:
1) $x\in (-\infty, N)$ then $|f(x)+5|<1$ implies that $|f(x)|<6$

2) $x\in (N,\infty)$ then $|f(x)-5|<1$ implies that $|f(x)|<6$

3) $x\in [-N, N]$ implies that $|f(x)|<B$
 
  • #9
Plato said:
You have three cases:
1) $x\in (-\infty, N)$ then $|f(x)+5|<1$ implies that $|f(x)|<6$

2) $x\in (N,\infty)$ then $|f(x)-5|<1$ implies that $|f(x)|<6$

3) $x\in [-N, N]$ implies that $|f(x)|<B$

Ok, I understand!

And could we also say $|f(x)|< \max \{6, B\}$ instead of $|f(x)|<B+6$ ?
 
  • #10
mathmari said:
And could we also say $|f(x)|< \max \{6, B\}$ instead of $|f(x)|<B+6$ ?

Well of course you can do do that, but why would you?
 
  • #11
Plato said:
Well of course you can do do that, but why would you?

Ok! I just wanted to know...

Thank you very much for your answer! :eek:
 
  • #12
It appears to me that all we can say is $M$ exists, we have no way of "finding a value for it" (such a function might have an arbitrarily large, but finite maximum, for example).
 

FAQ: Show that there is a M>0 such that |f(x)|<M

What does the notation |f(x)| mean in this statement?

The notation |f(x)| represents the absolute value of the function f(x). This means that regardless of the input value x, the output value of the function will always be positive.

Why is it important to prove that there exists a M>0 in this statement?

It is important to prove that there exists a M>0 because it guarantees that the function f(x) is bounded. This means that the values of the function do not grow infinitely large, making it easier to analyze and understand its behavior.

How does the value of M affect the statement?

The value of M determines the upper bound for the absolute value of the function |f(x)|. This means that for any input value x, the output value of the function will always be less than M in absolute value.

Can you provide an example of a function that satisfies this statement?

One example of a function that satisfies this statement is f(x) = sin(x). In this case, M=1 would be a suitable choice as |sin(x)| is always less than or equal to 1 for any input value of x.

Is it possible for the statement to hold true for all values of x?

Yes, it is possible for the statement to hold true for all values of x. This would mean that the function f(x) is a constant function, where the absolute value of the function is always less than some positive value M.

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