To evaluate the limit
, we can analyze the behavior of the expression as both
and
approach infinity.
Let's consider the numerator
and the denominator
separately.
For the numerator, as both
and
approach infinity, their sum
will also approach infinity.
For the denominator, we can rewrite it as
. As
and
approach infinity, the terms
and
will also approach infinity. Therefore, the denominator will also approach infinity.
Now, let's consider the entire fraction
. Since both the numerator and denominator approach infinity, we have an indeterminate form of
.
To evaluate this indeterminate form, we can apply techniques such as L'Hôpital's rule or algebraic manipulations. However, in this case, we can simplify the expression further.
By dividing both the numerator and denominator by
, we get:

As
approaches infinity, the terms
and
both approach zero. Similarly, the term
and
also approach zero.
Therefore, the limit simplifies to:

Hence, the limit
is equal to 0.