Question

limits

Answer 1

there are no limits lol

Answer 2

\[\lim_{n \rightarrow \infty}(1+\frac{ 5 }{ \sqrt n +3 })^{\sqrt n}=\lim_{n \rightarrow \infty}(1+\frac{1}{\frac{\sqrt n +3}{5}})^{\sqrt n}=\lim_{n \rightarrow \infty}(1+\frac{1}{\frac{\sqrt n +3}{5}})^{{\sqrt n}/5+3/5-3/5+{4\sqrt n}/5}\]

Answer 3

is this ok and what shoul I do now?

Answer 4

seems like you're on the right track already @gorica , this is called a fundamental limes.

Answer 5

oh snap im not good at algebra lol

Answer 6

I want to use formula \[\lim_{n \rightarrow \infty}(1+1/x)^x=e\]

Answer 7

Although in the process of your \(\LaTeX\) some of the informations seem to have got lost, I believe we both agree on the parenthesis, so we should first worry about with Exponent. there you should have something of the following:
\[\large \left(\frac{\sqrt{n}+3}{5}\right)^{\frac{5}{\sqrt{n}+3}^{\sqrt{n}} } \]
you agree?

Answer 8

why do I have to do that?

Answer 9

You want to bring your limit in the following form:
\[ \lim_{n \to \infty} \left(1 + \frac{1}{n} \right)^n=e\]

Answer 10

That is why, your limit is of the form \(1^\infty\) upon first inspection, whenever you have such a limit it is a good idea to choose that approach.

Answer 11

I wrote power as \[\frac{ \sqrt n }{ 5 }+\frac{3}{5}-\frac{3}{5}+\frac{4 \sqrt n}{5}\]

Answer 12

would you write it down? or tell me what can I do with what I've written if you know? :)

Answer 13

What you did I do not know, or to be honest, I don't know what it should be good for, but I can try to write out my approach, it will be a bit lengthy though. Keep in mind that you always want to use the fundamental limes which evaluates to e.

So you want to bring the following:
\[\large \lim_{n \to \infty} \left(1 + \frac{1}{\frac{\sqrt{n}+3}{5}} \right)^{\sqrt{n}} \] into the form of the fundamental limes:
\[\large \lim_{n \to \infty} \left( 1 + \frac{1}{n}\right)^n \]
How do you do that? Notice that your limes ALMOST looks like it, except that dreadful exponent \(\sqrt{n}\). To get rid of this it is simple, you just exponentiate your equation with the following:
\[\Large \left( \frac{\sqrt{n}+3}{5}\right)^{\frac{5}{\sqrt{n}+3}}=1 \]
What does that give you? the following:
\[\large \lim_{n \to \infty} \left(1 + \frac{1}{\frac{\sqrt{n}+3}{5}} \right)^{\left( \frac{\sqrt{n}+3}{5}\right)^{\frac{5}{\sqrt{n}+3}^{\sqrt{n}}}} \]
now take an intermediate step to evaluate what you have.

Answer 14

Can you already see where this is going @gorica ?

Answer 15

Obviously upon evaluate of the first two inner brackets you obtain the fundamental limes \(e\) it's in the exact same form, and you're left with:
\[\large \lim_{n \to \infty} e^{\frac{5\sqrt{n}}{\sqrt{n}+3}} \]

Answer 16

Because \[\large \lim_{n \to \infty} \left(1 + \frac{1}{\frac{\sqrt{n}+3}{5}} \right)^{\left( \frac{\sqrt{n}+3}{5}\right)}=e\]

Answer 17

I got it, thanks :)

Answer 18

very good.