I like to present totally cooked-up examples demonstrating the reason for certain hypotheses or limitations in calculus, in the hope that they will help students remember and correctly apply the theorems more easily. Here are a few.

• Note that the sequence has limit 0 but the function oscillates. This is why you can use convergence of the function to conclude convergence of the sequence of points along its graph, but not vice-versa. Intuitively, the function must “connect the dots” for its behavior to match the sequence’s (but the sequence can never be wilder than the function).
• Let a_n = n and b_n = -n. These sequences each diverge (to ) but their sum is constantly 0. That is why the Limit Laws only allow you to transfer convergence of individual sequences to their sums, products, etc., and not the reverse.
• Consider the series with terms 1, -1/2, 2/3, -1/3, 1/2, -1/4, 2/5, -1/5, … It is an alternating series whose terms limit to zero. If we sum consecutive pairs of terms, however, we obtain 1/2, 1/3, 1/4, 1/5, …: the harmonic series, which we know diverges. This is why the Alternating Series Test requires the magnitude of the terms decrease to zero.