Trigonometry with Fractional Part? Too Much!

Calculus Level 4

\[ \dfrac {\left ( \tan \left ( \{x\} - 1 \right ) \right ) \sin \{ x \} }{ \{ x \} \left ( \{ x \} - 1 \right ) } \]

What is the value of the above expression if we take for an arbitrary small value of \(x\)? Which is to say, what is the limit of the said expression when \(x \) approaches \(0\)?

Details and assumptions:

  • \( \{ A \} \) denote the fractional part of \(A\). For example, \(A = 1.74 \Rightarrow \{ A \} = 0.74 \), \( B = -1.3 \Rightarrow \{ B \} = 0.7 \)

  • \(\{x\}=x-\lfloor x \rfloor\)


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Image Credit: Wikipedia Fraction
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