Find $y$ ?

Calculus Level 3

$\lim_{x\to0 } \left [\left \lfloor \dfrac {\sin x}x \right \rfloor + \left \lfloor \dfrac x{\sin x} \right \rfloor + \left \lfloor \dfrac {\tan x}x \right \rfloor + \left \lfloor \dfrac x {\tan x} \right \rfloor + \lfloor \cos x \rfloor \right] = y + \dfrac1y$

Find the value of $y$ satisfying the equation above.

Notation : $\lfloor \cdot \rfloor$ denotes the floor function .

\frac { 3-\sqrt { 5 } }{ 2 } \quad

2+\sqrt { 3 } \quad

2-\sqrt { 3 } \quad

\frac { 3+\sqrt { 5 } }{ 2 } \quad

1

1 solution

Sanath Balaji
Jun 2, 2016

$Since,\\ sin\quad x\quad \le \quad x\quad \le \quad tan\quad x\quad and\quad \quad \underset { x\rightarrow 0 }{ lt\quad } cos\quad x\quad <\quad 1\\ y+\frac { 1 }{ y } =2\\ { y }^{ 2 }-2y+1=0\\ y=1$

Find y y y ?

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Find $y$ ?