What is limit of the function (x^5+1)/(x^7+1), x-->-1 ?

To evaluate the limit, we'll re-write the numerator and
denominator, using the identity:

a^n + b^n = (a+b)*(a^(n-1)
- a^(n-2)*b + a^(n-3)*b^2 - a^(n-4)*b^3 + ....)

According
to this formula, we'll get:

x^5 + 1 = (x + 1)(x^4 - x^3 +
x^2 - x + 1)

x^7 + 1 = (x + 1)(x^6 - x^5 + x^4 - x^3 + x^2
- x + 1)

(x^5+1)/(x^7+1) = (x^4 - x^3 + x^2 - x + 1)/(x^6 -
x^5 + x^4 - x^3 + x^2 - x + 1)

We'll evaluate the
limit:

lim (x^4 - x^3 + x^2 - x + 1)/(x^6 - x^5 + x^4 - x^3
+ x^2 - x + 1) = [(-1)^4-(-1)^3+ (-1)^2 +1  +
1]/(1+1+1+1+1+1+1)

lim (x^4 - x^3 + x^2 - x + 1)/(x^6 - x^5
+ x^4 - x^3 + x^2 - x + 1) =
(1+1+1+1+1)/(1+1+1+1+1+1+1)

lim (x^4 - x^3 + x^2 - x +
1)/(x^6 - x^5 + x^4 - x^3 + x^2 - x + 1) =
5/7

The value of the limit of the function,
if x approaches to -1, is: lim (x^5+1)/(x^7+1) =
5/7.