• Support PF! Buy your school textbooks, materials and every day products Here!

What is limx-->0 f(x) when lim x-->0 g(x)=0?

  • Thread starter lep11
  • Start date
  • #1
380
7

Homework Statement


Let |f(x)|≤g(x) for all x∈ Mf∩Mg.
What is limx-->0 f(x) when lim x-->0 g(x)=0?
What is limx-->0 f(x) when lim x-->0 g(x)=3?


The Attempt at a Solution


Well, given that|f(x)|≤g(x), lim x-->0 g(x)=0 intuitively implies to me that | limx-->0 f(x) |≤0
therefore | limx-->0 f(x) |=0 --> limx-->0 f(x)=0

AND when lim x-->0 g(x)=3

|limx-->0 f(x)|≤3 ⇔ -3 ≤limx-->0 f(x)≤3

Is my reasoning correct?
 

Answers and Replies

  • #2
pasmith
Homework Helper
1,748
420

Homework Statement


Let |f(x)|≤g(x) for all x∈ Mf∩Mg.
What is limx-->0 f(x) when lim x-->0 g(x)=0?
What is limx-->0 f(x) when lim x-->0 g(x)=3?


The Attempt at a Solution


Well, given that|f(x)|≤g(x), lim x-->0 g(x)=0 intuitively implies to me that | limx-->0 f(x) |≤0
Can you turn your intuition into a rigorous proof that [itex]\lim_{x \to 0} f(x) = 0[/itex] in this case?

AND when lim x-->0 g(x)=3

|limx-->0 f(x)|≤3 ⇔ -3 ≤limx-->0 f(x)≤3

Is my reasoning correct?
Must [itex]\lim_{x \to 0} f(x)[/itex] necessarily exist when [itex]\lim_{x \to 0} g(x) > 0[/itex]? Consider the case [tex]
f(x) = \begin{cases} -1 & x < 0, \\ 0 & x = 0, \\ 1 & x > 0, \end{cases}\qquad g(x) = 3.[/tex]
 
  • #3
380
7
Can you turn your intuition into a rigorous proof that [itex]\lim_{x \to 0} f(x) = 0[/itex] in this case?
How?

Must [itex]\lim_{x \to 0} f(x)[/itex] necessarily exist when [itex]\lim_{x \to 0} g(x) > 0[/itex]? Consider the case [tex]
f(x) = \begin{cases} -1 & x < 0, \\ 0 & x = 0, \\ 1 & x > 0, \end{cases}\qquad g(x) = 3.[/tex]
No. So in that case we cannot say anythin about [itex]\lim_{x \to 0} f(x) [/itex] since it won't necessarily even exist?

However if it exists then -3 ≤limx-->0 f(x)≤3
 
  • #4
380
7
ahh there is a small typo in the problem statement.. it should be x approaches a, not 0
 
  • #5
380
7
I used the squeeze theorem to prove that limx→0f(x)=0
 
  • #6
33,631
5,288
ahh there is a small typo in the problem statement.. it should be x approaches a, not 0
I used the squeeze theorem to prove that limx→0f(x)=0
Isn't it the limit as x approaches a?


So for the second part, if ##|f(x)| \le g(x)## and ##\lim_{x \to a} g(x) = 3##, is it necessarily true that ##\lim_{x \to a} f(x) = 0##?
 
  • #7
380
7
Isn't it the limit as x approaches a?
Yes it is, I'm sorry

So for the second part, if ##|f(x)| \le g(x)## and ##\lim_{x \to a} g(x) = 3##, is it necessarily true that ##\lim_{x \to a} f(x) = 0##?
No, because limx→a f(x) won't necessarily exist when lim x-->a g(x)=3.
 
  • #8
33,631
5,288
I used the squeeze theorem to prove that limx→0f(x)=0
So for the second part, if ##|f(x)| \le g(x)## and ##\lim_{x \to a} g(x) = 3##, is it necessarily true that ##\lim_{x \to a} f(x) = 0##?
No, because limx→a f(x) won't necessarily exist when lim x-->a g(x)=3.
OK, I wasn't sure whether you were using the squeeze theorem for the second part.
 

Related Threads on What is limx-->0 f(x) when lim x-->0 g(x)=0?

  • Last Post
Replies
3
Views
7K
  • Last Post
Replies
13
Views
4K
  • Last Post
Replies
2
Views
4K
Replies
3
Views
7K
Replies
11
Views
6K
  • Last Post
Replies
2
Views
21K
Replies
5
Views
3K
  • Last Post
Replies
2
Views
1K
Replies
1
Views
2K
Top