Limits Question

Question

\(f\left(x\right)=\left[\left(\frac{min\left(t^2+4t+6\right)\sin \left(x\right)}{x}\right)\right]\)

where [.] represents greatest integer function, then find \(\lim _{x\to 0}\left(f\left(x\right)\right)\)

Answer 1

f(x) = \([\frac{min(t^2+4t+6)sin x}x]\)

∵ t² + 4t + 6 = (t + 2)² + 2 \(\geq\) 2

∴ min (t² + 4t + 6) = 2

∴ f(x) = \([\frac{2sinx}x]\)

∵ sin x \(\leq\) x

⇒ -1 < \(\frac{sin x}x\leq1\)

⇒ -2 < \(\frac{2sin x}x\leq2\) 

∴ \([\frac{2sinx}x]\) = \(\begin{cases}2&;x=0or sinx/x=1\\1&;1/2 \leq\frac{sin x}x<1\\0&;0\leq\frac{sin x}x<1/2\\-1&;\frac{sin x}x<0\end{cases}\) 

\(\lim\limits_{x\to 0}f(x) = \lim\limits_{x\to 0}[\frac{2sin x}x]\) = 1