Question-51005

Question Number 51005 by Tawa1 last updated on 23/Dec/18

Commented by maxmathsup by imad last updated on 23/Dec/18

l e t A (θ) = \frac{s i n (θ - \frac{π}{6})}{\sqrt{3} - 2 c o s θ} c h a n g e m e n t θ - \frac{π}{6} = x g i v e

A (θ) = \frac{s i n x}{\sqrt{3} - 2 c o s (x + \frac{π}{6})} = \frac{s i n x}{\sqrt{3} - 2 (c o s x c o s (\frac{π}{6}) - s i n x s i n (\frac{π}{6}))}

= \frac{s i n x}{\sqrt{3} - 2 (\frac{\sqrt{3}}{2} c o s - \frac{1}{2} s i n x)} = \frac{s i n x}{\sqrt{3} - \sqrt{3} c o s x + s i n x} b u t {l i m}_{θ \to \frac{π}{6}} = {l i m}_{x \to 0} \frac{s i n x}{\sqrt{3} - \sqrt{3} c o s x + s i n x}

s i n x \sim x a n d c o s x \sim 1 - \frac{x^{2}}{2} \Rightarrow \frac{s i n x}{\sqrt{3} - \sqrt{3} c o s x - s i n x} \sim \frac{x}{\sqrt{3} - \sqrt{3} (1 - \frac{x^{2}}{2}) - x}

= \frac{x}{\frac{\sqrt{3}}{2} x^{2} - x} = \frac{1}{\frac{\sqrt{3}}{2} x - 1} \to - 1 (x \to 0) \Rightarrow {l i m}_{θ \to \frac{π}{6}} A (θ) = - 1 .

e r r o r o f t y p o \frac{s i n x}{\sqrt{3} - \sqrt{3} c o s x + s i n x} \sim \frac{x}{\sqrt{3} - \sqrt{3} (1 - \frac{x^{2}}{2}) + x} = \frac{x}{\frac{\sqrt{3}}{2} x^{2} + x} =

= \frac{1}{\frac{\sqrt{3}}{2} x + 1} \to 1 (x \to 0) \Rightarrow ★ {l i m}_{θ \to \frac{π}{6}} A (θ) = 1 ★

Commented by maxmathsup by imad last updated on 23/Dec/18

l e t A (x) = \frac{1 - c o s x \sqrt{c o s (2 x)}}{x^{2}} w e h a v e x \in V (0)

c o s x \sim 1 - \frac{x^{2}}{2} a n d c o s (2 x) \sim 1 - \frac{{(2 x)}^{2}}{2} = 1 - 2 x^{2} \Rightarrow \sqrt{c o s (2 x)} \sim \sqrt{1 - 2 x^{2}}

\sim 1 + \frac{1}{2} (- 2 x^{2}) = 1 - x^{2} \Rightarrow A (x) \sim \frac{1 - (1 - \frac{x^{2}}{2}) (1 - x^{2})}{x^{2}}

= \frac{1 - (1 - x^{2} - \frac{x^{2}}{2} + \frac{x^{4}}{2})}{x^{2}} = \frac{x^{2} + \frac{x^{2}}{2} - \frac{x^{4}}{2}}{x^{2}} = \frac{3}{2} - \frac{x^{2}}{2} \to \frac{3}{2} (x \to 0) \Rightarrow

{l i m}_{x \to 0} A (x) = \frac{3}{2} .

Commented by Tawa1 last updated on 24/Dec/18

God bless you sir

Answered by tanmay.chaudhury50@gmail.com last updated on 23/Dec/18

2) t = θ - \frac{π}{6}

l i \underset{t \to 0}{m} \frac{s i n (t)}{\sqrt{3} - 2 c o s (t + \frac{π}{6})}

= l i \underset{t \to 0}{m} \frac{s i n t}{\sqrt{3} - 2 (c o s t . \frac{\sqrt{3}}{2} - s i n t . \frac{1}{2})}

= l i \underset{t \to 0}{m} \frac{s i n t}{\sqrt{3} - \sqrt{3} c o s t + s i n t}

= l i \underset{t \to 0}{m} \frac{2 s i n \frac{t}{2} c o s \frac{t}{2}}{\sqrt{3} \times 2 {s i n}^{2} \frac{t}{2} + 2 s i n \frac{t}{2} c o s \frac{t}{2}}

= l i \underset{t \to 0}{m} \frac{c o s \frac{t}{2}}{\sqrt{3} s i n \frac{t}{2} + c o s \frac{t}{2}} = 1

o r m e t h o d

l i \underset{θ \to \frac{π}{6}}{m} \frac{s i n (θ - \frac{π}{6})}{2 [\frac{\sqrt{3}}{2} - c o s (θ)]} [\frac{\sqrt{3}}{2} = c o s \frac{π}{6}]

\frac{1}{2} l i \underset{θ \to \frac{π}{6}}{m} \frac{2 s i n (\frac{θ - \frac{π}{6}}{2}) c o s (\frac{θ - \frac{π}{6}}{2})}{2 s i n (\frac{\frac{π}{6} + θ}{2}) s i n (\frac{θ - \frac{π}{6}}{2})}

= \frac{1}{2} \times \frac{1}{\frac{1}{2}} = 1

Commented by Tawa1 last updated on 23/Dec/18

God bless you sir

Commented by tanmay.chaudhury50@gmail.com last updated on 23/Dec/18

m o s t w e l c o m e \dots

Answered by tanmay.chaudhury50@gmail.com last updated on 23/Dec/18

3) \lim_{x \to 0} \frac{1 - c o s x \sqrt{c o s 2 x}}{x^{2}}

\lim_{x \to 0} \frac{1 - {c o s}^{2} x (\frac{1 - {t a n}^{2} x}{1 + {t a n}^{2} x})}{x^{2} (1 + c o s x \sqrt{c o s 2 x})}

= \lim_{x \to 0} \frac{{s e c}^{2} x - {c o s}^{2} x + {s i n}^{2} x}{{s e c}^{2} x \times x^{2} \times (1 + c o s x \sqrt{c o s 2 x})}

= \frac{1}{2} \lim_{x \to 0} \frac{\frac{1}{{c o s}^{2} x} - {c o s}^{2} x + {s i n}^{2} x}{x^{2}}

\frac{1}{2} \lim_{x \to 0} \frac{{s i n}^{2} x . (1 + {c o s}^{2} x) + {s i n}^{2} x}{x^{2} {c o s}^{2} x}

= \frac{1}{2} \lim_{x \to 0} {(\frac{s i n x}{x})}^{2} \times (\frac{2 + {c o s}^{2} x}{{c o s}^{2} x})

= \frac{1}{2} \times 1 \times \frac{3}{2} = \frac{3}{4}

Commented by Tawa1 last updated on 23/Dec/18

God bless you sir

Answered by tanmay.chaudhury50@gmail.com last updated on 23/Dec/18

1) \lim_{y \to 0} \frac{x {s e c (x + y) - s e c x} + y s e c (x + y)}{y}

= x [\lim_{y \to 0} \frac{s e c (x + y) - s e c x}{y}] + s e c x

= x [\lim_{y \to 0} \frac{c o s x - c o s (x + y)}{y c o s x c o s (x + y)}] + s e c x

= x [\lim_{y \to 0} \frac{2 s i n (x + \frac{y}{2}) s i n (\frac{y}{2})}{c o s x c o s (x + y) \times \frac{y}{2} \times 2}] + s e c x

= x \times \frac{s i n x}{{c o s}^{2} x} + s e c x

= x t a n x s e c x + s e c x

Commented by Tawa1 last updated on 23/Dec/18

God bless you sir

Commented by tanmay.chaudhury50@gmail.com last updated on 23/Dec/18

m o s t w e l c o m e \dots