solutions of an inequation

[Vali]

If $\displaystyle a\in \mathbb{N}$ and $\displaystyle L(a)=\lim_{n \to \infty }\frac{1}{n}\int_{0}^{1}[ne^{ax}]dx$ then I need to find the solutions of $\displaystyle L(a)\leq e$
Right answer: {0,1}
Some ideas?

 

[Steven G]

I would solve such a simple integral. I would also factor out n (note that in this case |n| = n)

 

[Vali]

I just can't start.I don't know how.
For example L(0)=limit(n->00) 1/n*[n].If [n]=n then L(0)=1, right?
How to find for all a ?
Is it correct to take n out from floor function ? Like [nx]=n[x] ?
An off-topic question: For example if I have a limit from an integral.Is it correct to take n out from integral ? Like $\displaystyle \int 5nx=n\int 5x$

 

[tkhunny]

I just can't start.I don't know how.
For example L(0)=limit(n->00) 1/n*[n].If [n]=n then L(0)=1, right?
How to find for all a ?
Is it correct to take n out from floor function ? Like [nx]=n[x] ?
An off-topic question: For example if I have a limit from an integral.Is it correct to take n out from integral ? Like $\displaystyle \int 5nx=n\int 5x$

Are those square brackets a FLOOR and not Absolute Values? Well, that's a different problem. $\displaystyle \lfloor{ne^{ax}}\rfloor$

You can remove constants, even arbitrary constants. You can also remove things treated like constants, such as unused INDEPENDENT variables.

$$\int x\cdot y\;dx = y\cdot\int x\;dx$$
It is NOT appropriate to factor "n" from the floor function. floor(5 * 1.2) = floor(6) = 6 but floor(5 * 1.2) ==> 5*floor(1.2) = 5*1 = 5 and that is NOT 6. You can test these things with small examples. It isn't a great way to PROVE that things work, but it can be exceptional at proving that things DON'T work.

 

[Vali]

I solved the exercise.I used x-1<= [x] <= x
Thank you for your help

 

[Steven G]

I would solve such a simple integral. I would also factor out n (note that in this case |n| = n)

Oh, you meant the greatest integer function. My eye sight is starting to fail!

 

[MarkFL]

Oh, you meant the greatest integer function. My eye sight is starting to fail!

I wasn't sure initially what the brackets meant, and typically when the greatest integer function is intended, this is explicitly stated.