Question

Build a table to estimate lim x->0 sin(1/x) *use multiple of pi

Answer 1

Hmm, I would suggest using multiples of \(\dfrac{1}{\pi}\) instead, unless you have a calculator to compute things like \(\sin\dfrac{1}{10\pi}\).

For example, taking \(x=\dfrac{2}{n\pi}\) with \(n=1,2,3,\ldots\) you have the following table:
\[\begin{array}{c|ccccccc}
x&\frac{1}{\pi}&\frac{2}{3\pi}&\frac{1}{2\pi}&\frac{2}{5\pi}&\frac{1}{4\pi}&\frac{2}{7\pi}&\frac{1}{8\pi}\\[1ex]
\hline
\sin\frac{1}{x}&\sin\pi&\sin\frac{3\pi}{2}&\sin2\pi&\sin\frac{5\pi}{2}&\sin4\pi&\sin\frac{7\pi}{2}&\sin8\pi
\end{array}\]Notice that as \(n\to\infty\), you have \(x\to0\), so the values of the function in the second row should give you some indication as to the value of the desired limit if it exists, or some indication as to why it might not exist.

Answer 2

oh okay so okay I slowly am starting to understand, but thank you for the extensive explanation. It really helps me fully understand how to break down a problem like this one.

Answer 3

I am sure you were supposed to use a calculator on this.

So plug in numbers very close to \(0\) on the left and the right.

Plug in 0.001,0.1,0.0001,0.010101, 0.000000001 and then do the same numbers but negative.