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I apologize for the lengthy question in advance. As you can see in the title, I'm trying to calculate the limits of the two functions.

They are defined the following:

(1) If $\lim_{x\to 0}f(x) = 2$ , what is $\lim_{x\to 0}(f(x)+f(5x))$ ?

(2) If $\lim_{x\to 0}(f(x)+1/f(x)) = 2$, what is $\lim_{x\to 0}f(x)$?

I've never worked with limits like these before. But these are my thoughts so far:

We know that $\lim_{x\to 0}f(x) = 2$ and since $\lim_{x\to 0}(f(x)+f(5x))$ = $\lim_{x\to 0}f(x)$ + $\lim_{x\to 0}f(5x)$ (At least I hope this is allowed), this results in $2$ + $\lim_{x\to 0}f(5x)$

However, I do not quite know how to go on from this. Is there a way I can manipulate $\lim_{x\to 0}f(5x)$ so I can use the given property to calculate a limit?

For (2), it's quite similar.I'm not quite sure if I can rewrite it again to $\lim_{x\to 0}f(x) + \lim_{x\to 0}1/f(x)$. I'm not sure if this is relevant to this task, but we're taking a limit of a function and it's inverse (not the inverse function, but the inverse of the value). Is this something I can use for this exercise or is this just an unnecessary observation in this case?

23408924
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  • I will give you a hint for 1) but I strongly suggest to read more about limits. Those are some pretty basic stuff. Anyway, here goes the hint: set $ 5x = y$. – cgss Nov 02 '20 at 09:02

1 Answers1

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For the first one we can use that

$$\lim_{x\to 0} 5x \to 0$$

and then by $y=5x \to 0$

$$\lim_{x\to 0}f(5x)=\lim_{y\to 0}f(y) =0$$

by this theorem

For the second one refer to

user
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  • Thank you for your answer! I'll check it out. How do we know that $\lim_{x\to 0}5x = 0$? – 23408924 Nov 02 '20 at 09:27
  • @23408924 This is a basic limit, by multiplication rule $\lim_{x\to 0}5x =\lim_{x\to 0}5 \cdot \lim_{x\to 0} x =5 \cdot 0=0$. If you are just starting to learn with limits the second one is not trivial at all. – user Nov 02 '20 at 09:29
  • Thanks. You're right. I should know this I guess I just completely forgot. – 23408924 Nov 02 '20 at 09:38
  • @23408924 This is not an issue, this site is for math lovers at any level, any contribution is important and you can find a lot of material here, just search it and ask when something is not completely clear! Bye – user Nov 02 '20 at 09:40