STATEMENT-1: limn→∞(1.2n3+2n2+1+2.3n3+2n2+2+3.4n3+2n2+3+−−−+1n+1) is equal to 13STATEMENT-2: Let f,g,h be three functions such that f(x)≤g(x)≤h(x) for all x in same interval containing the point x=c and if limx→cf(x)=limx→ch(x)=l then limx→cg(x)=l

1.2+2.3+3.4+...+n(n+1) =∑n2+n=n(n+1)(2n+1)6+n(n+1)2 =n(n+1)(n+2)3Let the Given problem be P limn→∞n(n+1)(n+2)3(n3+2n2+n)≤P≤limn→∞n(n+1)(n+2)3(n3+2n2+1)

STATEMENT-1: limn→∞(1.2n3+2n2+1+2.3n3+2n2+2+3.4n3+2n2+3+−−−+1n+1) is equal to 13STATEMENT-2: Let f,g,h be three functions such that f(x)≤g(x)≤h(x) for all x in same interval containing the point x=c and if limx→cf(x)=limx→ch(x)=l then limx→cg(x)=l

1.2+2.3+3.4+...+n(n+1) =∑n2+n=n(n+1)(2n+1)6+n(n+1)2 =n(n+1)(n+2)3Let the Given problem be P limn→∞n(n+1)(n+2)3(n3+2n2+n)≤P≤limn→∞n(n+1)(n+2)3(n3+2n2+1)

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STATEMENT-1: $\lim_{n \to \infty} (\frac{1.2}{n^{3} + 2 n^{2} + 1} + \frac{2.3}{n^{3} + 2 n^{2} + 2} + \frac{3.4}{n^{3} + 2 n^{2} + 3} + - - - + \frac{1}{n + 1})$ is equal to $\frac{1}{3}$
STATEMENT-2: Let f,g,h be three functions such that $f (x) \leq g (x) \leq h (x)$ for all x in same interval containing the point $x = c$ and if $\lim_{x \to c} f (x) = \lim_{x \to c} h (x) = l$ then $\lim_{x \to c} g (x) = l$

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Statement-1 is false, statement-2 is true

Statement-1 is true, statement-2 is false

Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1

Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1

answer is A.

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Detailed Solution

$1.2 + 2.3 + 3.4 + ... + n (n + 1) = \sum n^{2} + n = \frac{n (n + 1) (2 n + 1)}{6} + \frac{n (n + 1)}{2} = \frac{n (n + 1) (n + 2)}{3}$

Let the Given problem be P

$\lim_{n \to \infty} \frac{n (n + 1) (n + 2)}{3 (n^{3} + 2 n^{2} + n)} \leq P \leq \lim_{n \to \infty} \frac{n (n + 1) (n + 2)}{3 (n^{3} + 2 n^{2} + 1)}$