Find-the-largest-value-of-the-non-negative-integer-p-for-which-lim-x-1-px-sin-x-1-p-x-sin-x-1-1-1-x-1-x-1-4-

Question Number 216077 by MATHEMATICSAM last updated on 27/Jan/25

Find the largest value of the non negative

integer p for which

\lim_{x \to 1} {\frac{- p x + \sin (x - 1) + p}{x + \sin (x - 1) - 1}}^{\frac{1 - x}{1 - \sqrt{x}}} = \frac{1}{4} .

Answered by mahdipoor last updated on 27/Jan/25

l n (y) = (\frac{1 - x}{1 - \sqrt{x}}) l n (\frac{s i n (x - 1) + p (x - 1)}{s i n (x - 1) + (x - 1)})

\Rightarrow \lim_{x \to 1} \frac{1 - x}{1 - \sqrt{x}} = 1 + \sqrt{x} = 2

\Rightarrow \lim_{x \to 1} \frac{s i n (x - 1) + p (x - 1)}{s i n (x - 1) + (x - 1)} = \frac{0}{0} \Rightarrow h o p i t a l \Rightarrow

\Rightarrow = \frac{c o s (x - 1) + p}{c o s (x - 1) + 1} = \frac{p + 1}{2}

\lim_{x \to 1} l n (y) = l n (\frac{1}{4}) = 2 \times l n (\frac{p + 1}{2})

\Rightarrow p = 0