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How many solutions does $\tan(\frac{1}{x}) = 0$ have on the interval $-1 \leq x \leq 1$?

I do not know how to get the coefficient of $x$ here. I put this into a graphing calculator and it gave me a straight line. So I am assuming the answer to the question is infinite because $x$ could be any one of the infinite points here. I just don't understand how to get the coefficient and/or how to find $x$ in this problem. How do I cancel out $\tan()$, so I can find what $x$ is?

user93089
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KKendall
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1 Answers1

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Recall that $\tan(y) = 0 \implies y= n\pi$, where $n \in \mathbb{Z}$. Hence, $\tan(1/x) = 0 \implies x = \dfrac1{n \pi}$, where $n \in \mathbb{Z} \backslash \{0\}$. Now what can you say about $\dfrac1{n \pi}$, where $n \in \mathbb{Z} \backslash \{0\}$? Are they all in the interval $[-1,1]$?

  • They all must be right? Because the highest value x can be is $\dfrac1{\pi}$ which is around .3. And it will only get smaller; $\dfrac1{20 \pi}$ is tiny. – KKendall Sep 25 '13 at 03:25
  • @KKendall Yes... –  Sep 25 '13 at 03:36