Let f:R→R be defined asfx=limn→∞x1+n2+3x2+n2+5x3+n2+.....+2n-1xn+n2; x≠π21when x=π2 (Where [.] denotes greatest integer function), then which of following statement (s) is/are correct?

By using sandwiteh theorem n 2 n + n 2 < 1 1 + n 2 + 3 2 + n 2 + . . . + 2 n - 1 n + n 2 < n 2 1 + n 2 ∴ lim n → ∞ 1 1 + n 2 + 3 2 + n 2 + . . . + 2 n - 1 n + n 2 = 1 Hence f x x ; x ≠ π 2 1 ; x = π 2

f x is non - derivable at x = π 2

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Let $f : R \to R$ be defined as
$f (x) =$ $\{\begin{cases} \lim_{n \to \infty} (\frac{[x]}{1 + n^{2}} + \frac{3 [x]}{2 + n^{2}} + \frac{5 [x]}{3 + n^{2}} + . . . . . + \frac{(2 n - 1) [x]}{n + n^{2}}) & ; x \neq \frac{π}{2} \\ 1 & when x = \frac{π}{2} \end{cases}$ (Where [.] denotes greatest integer function), then which of following statement (s) is/are correct?

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$f (x) is non - derivable at x = \frac{π}{2}$

$f (x) is discontinuous at all integers and continuous at x = \frac{π}{2}$

$f (x) is unbounded function .$

$f (x) is injective but not surjective$

answer is C, D.

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Detailed Solution

By using sandwiteh theorem

$\frac{n^{2}}{n + n^{2}} < \frac{1}{1 + n^{2}} + \frac{3}{2 + n^{2}} + . . . + \frac{2 n - 1}{n + n^{2}} < \frac{n^{2}}{1 + n^{2}} ∴ \lim_{n \to \infty} \{\frac{1}{1 + n^{2}} + \frac{3}{2 + n^{2}} + . . . + \frac{2 n - 1}{n + n^{2}}\} = 1 Hence f (x) \{\begin{matrix} [x]; x \neq \frac{π}{2} \\ 1; x = \frac{π}{2} \end{matrix}\}$