InterviewSolution
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InterviewSolution
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\(f\left(x\right)=\left[\left(\frac{min\left(t^2+4t+6\right)\sin \left(x\right)}{x}\right)\right]\)where [.] represents greatest integer function, then find \(\lim _{x\to 0}\left(f\left(x\right)\right)\) |
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Answer» 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 |
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