Determine whether the sequence
Using the limit laws for infinite sequence, we would evaluate
Because our limit evaluates to a finite number, the sequence converges (and it converges to 1/2).
The above example is trivial, but demonstrates why we need the limit laws. They allow us to evaluate limits of more complicated sequences.
Explanation of Each Step
We applied the definition of convergence of a sequence. Recall that to determine if a sequence is convergent we evaluate
and if this limit exists, the sequence converges. If it doesn't the sequence is divergent.
To make the limit easier to evaluate, we divided both the numerator and denominator by . This is as commonly used trick when evaluating limits.
Steps (4, 5)
These two steps consisted of evaluating our limits and simple algebraic manipulation and should be straightforward to students who are familiar with limits of functions.
Memorizing the Limit Laws
Students are expected to memorize the laws, but not the labels we assigned to them. For example, students are expected to have memorized that
but do not need to memorize that this is the third law.