Let f(n)= displaystyle lim x arrow 0((1+ sin (x/2))(1+ sin (x/22)) ldots .(1+ sin (x/2n)))1 / x . If displaystyle lim n arrow ∞ f(n)=(3 e/k) then find k

Question

Let f(n)= displaystyle lim x arrow 0((1+ sin (x/2))(1+ sin (x/22)) ldots .(1+ sin (x/2n)))1 / x . If displaystyle lim n arrow ∞ f(n)=(3 e/k) then find k (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

f(n)= displaystyle lim x arrow 0(1+ sin (x/2))1 / x(1+ sin (x/22))1 / x ⋅s ⋅s ⋅s(1+ sin (x/2n))1 / x =e(1/2) ⋅ e(1/22) ldots ldots . e(1/2n) Now, displaystyle lim n arrow ∞ f(n)=e(1 / 2/1-1 / 2)=e

Q. Let $f (n) = x \to 0 lim ((1 + sin \frac{x}{2}) (1 + sin \frac{x}{2 ^{2}}) \dots . (1 + sin \frac{x}{2 ^{n}}))^{1/ x} .$ If $n \to \infty lim f (n) = \frac{3 e}{k}$ then find $k$

$f (n) = x \to 0 lim (1 + sin \frac{x}{2})^{1/ x} (1 + sin \frac{x}{2 ^{2}})^{1/ x}$
$\dots\dots\dots (1 + sin \frac{x}{2 ^{n}})^{1/ x}$
$= e^{\frac{1}{2}} \cdot e^{\frac{1}{2 ^{2}}} \dots\dots . e^{\frac{1}{2 ^{n}}}$
Now, $n \to \infty lim f (n) = e^{\frac{1/2}{1 - 1/2}} = e$

Q. Let f(n)=x→0lim​((1+sin2x​)(1+sin22x​)….(1+sin2nx​))1/x. If n→∞lim​f(n)=k3e​ then find k

f(n)=x→0lim​(1+sin2x​)1/x(1+sin22x​)1/x ⋯⋯⋯(1+sin2nx​)1/x =e21​⋅e221​…….e2n1​ Now, n→∞lim​f(n)=e1−1/21/2​=e

Q. Let $f (n) = x \to 0 lim ((1 + sin \frac{x}{2}) (1 + sin \frac{x}{2 ^{2}}) \dots . (1 + sin \frac{x}{2 ^{n}}))^{1/ x} .$ If $n \to \infty lim f (n) = \frac{3 e}{k}$ then find $k$

$f (n) = x \to 0 lim (1 + sin \frac{x}{2})^{1/ x} (1 + sin \frac{x}{2 ^{2}})^{1/ x}$
$\dots\dots\dots (1 + sin \frac{x}{2 ^{n}})^{1/ x}$
$= e^{\frac{1}{2}} \cdot e^{\frac{1}{2 ^{2}}} \dots\dots . e^{\frac{1}{2 ^{n}}}$
Now, $n \to \infty lim f (n) = e^{\frac{1/2}{1 - 1/2}} = e$