Prove direct from definition

[asset101]

let an = $$\frac{(2n+1)}{(3n+7)}$$. Prove direct from definition that an$$\rightarrow2/3$$.

Any help would be appreciated.
Cheers

 

[tiny-tim]

ok … first step in a question like this is …

what is the definition of the statement "lim((2n+ 1)/(3n+7)) = 2/3"? ​

 

[AUMathTutor]

Here's an idea... as long as n isn't zero, the following two things are the same:

(2n+1)/(3n+7) = [(2n)(1+1/2n)]/[(3n)(1+7/3n)]

What happens as the n's get bigger and bigger to the terms 1/2n and 7/3n?

 

[HallsofIvy]

Here's an idea... as long as n isn't zero, the following two things are the same:

(2n+1)/(3n+7) = [(2n)(1+1/2n)]/[(3n)(1+7/3n)]

What happens as the n's get bigger and bigger to the terms 1/2n and 7/3n?

That's the simplest way of finding the limit but it doesn't "prove from the definition".