(1) lim_(x→1) (x^2 −1)^3 sin ((1/(x−1)))^3 ? (2) lim_(x→∞) ((x^2 (1+sin^2 x))/((x+sin x)^2 )) ?


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Question Number 122130 by bemath last updated on 14/Nov/20

$(1) \lim_{x \to 1} {(x^{2} - 1)}^{3} \sin {(\frac{1}{x - 1})}^{3} ?$ $(2) \lim_{x \to \infty} \frac{x^{2} (1 + \sin^{2} x)}{{(x + \sin x)}^{2}} ?$
	Answered by TANMAY PANACEA last updated on 14/Nov/20

	$1) 1 ⩾ s i n (\frac{1}{x - 1}) ⩾ - 1$ $s o \lim_{x \to 1} {(x^{2} - 1)}^{3} s i n {(\frac{1}{x - 1})}^{3}$ $= 0 \times (a n y v a l u e i n b e t w e e n \pm 1) = 0$
	Answered by Bird last updated on 14/Nov/20
	$2)f(x)=((x^2 (1+sin^2 x))/((x+sinx)^2 )) ⇒ f(x)=((x^2 +x^2 sin^2 x)/(x^2 +2xsinx +sin^2 x)) =((x^2 (1+sin^2 x))/(x^2 (1+((2sinx)/x)+((sin^2 x)/x^2 )))) =((1+sin^2 x)/(1+((2sinx)/x)+((sin^2 x)/x^2 ))) ⇒ lim_(x→∞) f(x)=lim_(x→∞) 1+sin^2 x =lim_(x→+∞) 1+((1−cos2x)/2) =lim_(x→∞) (1/2){3−cos(2x)} but cos(2x) havent any limit at∞ ⇒lim f(x) dont exist !$
	$2) f (x) = \frac{x^{2} (1 + {s i n}^{2} x)}{{(x + s i n x)}^{2}} \Rightarrow$ $f (x) = \frac{x^{2} + x^{2} {s i n}^{2} x}{x^{2} + 2 x s i n x + {s i n}^{2} x}$ $= \frac{x^{2} (1 + {s i n}^{2} x)}{x^{2} (1 + \frac{2 s i n x}{x} + \frac{{s i n}^{2} x}{x^{2}})}$ $= \frac{1 + {s i n}^{2} x}{1 + \frac{2 s i n x}{x} + \frac{{s i n}^{2} x}{x^{2}}} \Rightarrow$ ${l i m}_{x \to \infty} f (x) = {l i m}_{x \to \infty} 1 + {s i n}^{2} x$ $= {l i m}_{x \to + \infty} 1 + \frac{1 - c o s 2 x}{2}$ $= {l i m}_{x \to \infty} \frac{1}{2} {3 - c o s (2 x)}$ $b u t c o s (2 x) h a v e n t a n y l i m i t a t \infty$ $\Rightarrow l i m f (x) d o n t e x i s t!$
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