I just started my lesson on my elementary limit recently, and I did many exercises to consolidate my knowledge.

I figured out something interesting but I'm not sure whether it is correct.

That's why I need your superb help to prove or to reject it.

There are several ideas of mine.

1.[Math]\lim_{f(x)\to 0} \frac{1}{f(x)} = 0[/Math]

\lim_{f(x)\to\0} \frac{1}{f(x)} = 0

My idea: The limit does not exist.

I don't know how to prove it, because if the function is a fraction, I've heard of inverse function. But I have no idea to prove my thought.

2.[Math]\lim_{x\to -2} \frac{1}{[\sqrt(x+2)]} = 0[/Math]

The limit does not exist.

3.[Math]\lim_{x\to 1} \sqrt{x-1} = 0[/Math]

\lim_{x\to\1} \sqrt{x-1} = 0

My idea: It does not exist because we can't get the limit from the left.

Bolded words refer to non-latex .

I think you're right on all three counts. The easiest way of proving (1) is by providing a simple counter-example to show that it's not true for all functions f. As for (2) and (3), they're fairly simple. Just show that as x tends to the limit, the function becomes unbounded.

 

I've also corrected your LaTeX errors. The problem was that you were using \lim_{x\to\0} - the error is highlighted. You just need to use something like \lim_{x \to 0} instead.

Thank you for replacing with the correct ones. I used a lot of time to doing so but I wasn't able to fix it.

(1) is by providing a simple counter-example to show that it's not true for all functions

For (1), so it would be correct if f(x) is not a fraction?

i.e. [Math]f(x)=a_{n}x^n+a_{(n-1)}x^{(n-1)}+....+c=0[/Math]

For (2), what about 2=x where c is a real no.?

How can we prove that it cannot be further simplified ?

I'm not entirely sure about that first one. I thought it said:

 

[math]\lim_{x\to 0} \frac{1}{f(x)} = 0[/math]

 

But [imath]\lim_{f(x) \to 0} \frac{1}{f(x)} = 0[/imath] is just like saying [imath]\lim_{y \to 0} \frac{1}{y} = 0[/imath] - this obviously isn't true, in fact the limit doesn't even exist at that point.