18. Sequences

b2. Limits From Plots

One can often use a plot to guess the limit of a sequence.

The first \(30\) terms of the sequence \(a_n=\dfrac{n}{2n+(-1)^n}\) are plotted at the right. From the plot we expect that \(\lim\limits_{n\to\infty}a_n =\lim\limits_{n\to\infty}\dfrac{n}{2n+(-1)^n} =\dfrac{1}{2}\)

The plot shows a sequence of points denoted by squares. The first term
has a value of 0.4, and the rest of the terms alternate between being above and
below 0.5, forming two curves that approach 0.5 as n approaches infinity.

Use a plot to estimate the limit of the sequence \(a_n=\dfrac{3n}{2n+\sin n}\).

The plot of \(a_n=\dfrac{3n}{2n+\sin n}\) is shown at the right. So we expect \(\lim\limits_{n\to\infty}\dfrac{3n}{2n+\sin n} =\dfrac{3}{2}\).

The plot shows a sequence of points denoted by plus signs. The points
form a sinusodal curve that decreases in amplitude. The value of the terms
approach 1.5 as n approaches infinity.

A plot is not usually regarded as sufficient to show that a sequence has a limit nor to determine the value of the limit.

18. Sequences

b2. Limits From Plots

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