Let ln = (2n + (-2)n /2n) and Ln = (2n + (- 2)n/3n) then as n →∞

Question

Let ln = (2n + (-2)n /2n) and Ln = (2n + (- 2)n/3n) then as n →∞ (A) Both the sequences have limits (B)  displaystyle limn → ∞ ln exists but displaystyle limn → ∞ Ln does not exist (C)  displaystyle limn → ∞ ln does not exist but displaystyle limn → ∞ Ln exists (D) Both the sequences do not have limits.

Tardigrade Admin · Accepted Answer

Given, ln =(2n+(-2)n/2n)=1+((-2)n/2n) ...(i) And Ln=(2n+(-2)2 n/3n) ...(ii) Now, from Eq. (i), we get displaystyle lim n arrow ∞ ln = begincases0, text when n text is odd 2, text when n text is even endcases ∴ displaystyle lim n arrow ∞ does not exist and displaystyle lim n arrow ∞ Ln=0 ∴ displaystyle lim x arrow ∞ Ln exist.

Q. Let $l_{n} = \frac{2 ^{n} + ( - 2 ) ^{n}}{2 ^{n}}$ and $L_{n} = \frac{2 ^{n} + ( - 2 ) ^{n}}{3 ^{n}}$ then as $n \to \infty$

Both the sequences have limits

$n \to \infty lim l_{n}$ exists but $n \to \infty lim L_{n}$ does not exist

$n \to \infty lim l_{n}$ does not exist but $n \to \infty lim L_{n}$ exists

Both the sequences do not have limits.

Q. Let ln​=2n2n+(−2)n​ and Ln​=3n2n+(−2)n​ then as n→∞

Both the sequences have limits

n→∞lim​ln​ exists but n→∞lim​Ln​ does not exist

n→∞lim​ln​ does not exist but n→∞lim​Ln​ exists

Both the sequences do not have limits.

Given, ln =2n2n+(−2)n​=1+2n(−2)n​.˙.(i) And Ln​=3n2n+(−2)2n​...(ii) Now, from Eq. (i), we get n→∞lim​ ln ={0,2,​ when n is odd when n is even ​ ∴n→∞lim​ does not exist and n→∞lim​Ln​=0 ∴x→∞lim​Ln​ exist.

Q. Let $l_{n} = \frac{2 ^{n} + ( - 2 ) ^{n}}{2 ^{n}}$ and $L_{n} = \frac{2 ^{n} + ( - 2 ) ^{n}}{3 ^{n}}$ then as $n \to \infty$

$n \to \infty lim l_{n}$ exists but $n \to \infty lim L_{n}$ does not exist

$n \to \infty lim l_{n}$ does not exist but $n \to \infty lim L_{n}$ exists