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

## 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?

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pasmith
Homework Helper

## 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 $\lim_{x \to 0} f(x) = 0$ 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 $\lim_{x \to 0} f(x)$ necessarily exist when $\lim_{x \to 0} g(x) > 0$? Consider the case $$f(x) = \begin{cases} -1 & x < 0, \\ 0 & x = 0, \\ 1 & x > 0, \end{cases}\qquad g(x) = 3.$$

Can you turn your intuition into a rigorous proof that $\lim_{x \to 0} f(x) = 0$ in this case?
How?

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

However if it exists then -3 ≤limx-->0 f(x)≤3

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

Mark44
Mentor
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##?

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.

Mark44
Mentor
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.