# Proving a limit to infinity using epsilon-delta

#### brooklysuse

##### New member
lim 2x + 3 = ∞.
x→∞

Pretty intuitive when considering the graph of the function. But how would I show this using the epsilon-delta definition?

Thanks!

#### Evgeny.Makarov

##### Well-known member
MHB Math Scholar
Please expand the definition of the limit and write what you need to prove.

#### brooklysuse

##### New member
Looking to use this definition. f:A->R, A is a subset of R, (a, infinity) is a subset of A.

lim f(x) =infinity if for any d in R, there exists a k>a such that when x>k, then f(x)>d.
x->infinity

#### Evgeny.Makarov

##### Well-known member
MHB Math Scholar
You may forget about $a$. So you have to prove that for every $d$ there exists a $k$ such that if $x>k$, then $f(x)=2x+3>d$. So consider an arbitrary $d$. You need to show that there exists a $k$ such that $x>k$ implies $x>(d-3)/2$. It's sufficient to take $k=(d-3)/2$.