Evaluate the limit of the function [(x+5)/(x+2)]^(2x+1) x approaches to + infinite.

The limit will go to the base and to the
superscript.

The limit of the base
is:

lim (x+5)/(x+2) = lim x(1 + 5/x)/x(1 + 2/x) = lim(1 +
5/x)/(1 + 2/x)=1

The limit of superscript
is:

lim (2x+1) = +infinite

We
notice that we've get an indeterminacy: 1^infinite.

We'll
create remarcable limit "e".

We'll add 1 and subtract 1
inside the brackets of the base:

[1 + (x+5)/(x+2) - 1] = [1
+ (x+5-x-2)/(x+2)] = [1+ 3/(x+2)]

The remarcable limit
is:

lim (1 + 1/x)^x = e,
x->infinite

We'll re-write the
limit:

lim { [1+ 3/(x+2)]^(x+2)/3}^3(2x+1)/(x+2) =
lim { [1+ 3/(x+2)]^(x+2)/3}^ lim 3(2x+1)/(x+2)

lim { [1+
3/(x+2)]^(x+2)/3} = e

The limit of
superscript:

lim 3(2x+1)/(x+2) = lim 3x(2+1/x)/x(1+2/x)
=  lim 3(2+1/x)/(1+2/x) = 6/1

The limit of
the function, when x approaches to +infinite, is:  lim [(x+5)/(x+2)]^(2x+1) =
e^6