It often happens that we have a difference of two expressions that we would love to cancel and they would even cancel nicely, but they are hidden under roots so they cannot be put together.
Example:
How do we get rid of these roots?
Solution: Multiply and divide this difference by the same expression, but with plus, and use the formula
In the above example we would write
Similarly we can use algebraic identities to get rid of other roots. For instance, the identity
allows us to get rid of cubic roots similarly to the above example (note that n is also the cubic root of n3).
Example:
This was the most typical case; when looking for a limit, the only difference of roots that really troubles us is the "infinity minus infinity" type. Note that for such types there are other methods. First of all, there is the appropriate box for indeterminate difference, and if there are powers involved, one can also try to use the approach from the box polynomials and ratios with powers. The rule is simple. If there is just one dominant term (that is, if the two roots are of different types), we are better off using the box on polynomials, solving the problem using the scale of powers and factoring out. If both roots are of the same type, then the scale of powers won't help and this box is the best way out.
Note that our examples were appropriate; in the first example, each root is of the type n2; in the second type we have two expression of type n.
In Solved Problems - Limits, these methods are used in this problem.