How to treat limits such as these properly?

Also with this as well:

I see the limit is 13/23 since sinx~x for small x so the limit is 13/23 but I don't know how to write down a proper evaluation.

Say we have some limit.

$\displaystyle \lim_{x\rightarrow \infty }(\sqrt{x+10}-\sqrt x~)$

and I need to evaluate this limit.

What is the best way to do this obviously showing my working I know that sqrt(x+10) and sqrt(x) both tend to infinity and so the limit of function should be the limits of both parts taken away which is "infinity minus infinity" which is 0. I'm not sure whether this is a valid method and if it isn't what should I be doing?

Any help?

Thanks

Also, try multiplying the numerator and denominator of the argument of the limit by its conjugate, and see where that gets you. This is the best method I can come up with having put some thought into it!

Thanks, I get the idea I just want to make sure my work is "acceptable".

Also with this as well:

I see the limit is 13/23 since sinx~x for small x so the limit is 13/23 but I don't know how to write down a proper evaluation.

Well infinity minus infinity = 0 is definitely not true, and is probably unacceptable.

My first suggestion is probably slightly shaky, because although it tries to subtract the same limit from itself, I don't know if that is rigorous enough.

The multiplication thing is guaranteed to work because you end up with 10/infinity which is definitely zero.

You know that if the limit of $\displaystyle \lim_{x \rightarrow a}f$ exists, then you know the limit of $\displaystyle \lim_{x \rightarrow a}\dfrac{1}{f}$ also exists.

So $\displaystyle \dfrac{sin(13x)}{sin(23x)} = \dfrac{sin(13x)}{13x}\dfrac{23x}{sin(23x)}\dfrac{13x}{23x}$. So the limit must be 13/23.



But infinity - infinity can equal zero. i.e $\displaystyle \lim_{x \rightarrow \infty} x - x$.

OP can just do standard epsilon-delta symbol shunting.

$|(\sqrt{x + 10} - \sqrt{x}) - 0| = \sqrt{x + 10} - \sqrt{x} < \epsilon$.

Then rearrange to get $\displaystyle \left(\dfrac{10 - \epsilon^2}{2\epsilon}\right)^2 < x$. Now you can cook up the usual epsilon-delta proof.

Say we have some limit.

$\displaystyle \lim_{x\rightarrow \infty }(\sqrt{x+10}-\sqrt x~)$

and I need to evaluate this limit.

What is the best way to do this obviously showing my working I know that sqrt(x+10) and sqrt(x) both tend to infinity and so the limit of function should be the limits of both parts taken away which is "infinity minus infinity" which is 0. I'm not sure whether this is a valid method and if it isn't what should I be doing?

Any help?

Thanks