Question

The values of A and B such that the function

$a = 1,b = - 3$, is continuous everywhere are

• A. $a = 2,b = - 1$
• B. $\lim_{n \rightarrow \infty}\left\lbrack \frac{\sum n^{2}}{n^{3}} \right\rbrack =$
• C. $- \frac{1}{6}$
• D. $\frac{1}{6}$

Answer 1

Answer: (C)

$- \frac{1}{6}$

Answer 2

For continuity at all we must have

$f \left( - \frac { \pi } { 2 } \right) = \lim _ { x \rightarrow ( - \pi / 2 ) ^ { - } } ( - 2 \sin x )$ $= \lim _ { x \rightarrow ( - \pi / 2 ) ^ { + } } ( A \sin x + B )$

⇒ $2 = - A + B$ …..(i)

and $f \left( \frac { \pi } { 2 } \right) = \lim _ { x \rightarrow ( \pi / 2 ) ^ { - } } ( A \sin x + B )$ $= \lim _ { x \rightarrow ( \pi / 2 ) ^ { + } } ( \cos x )$

⇒ $0 = A + B$ ….(ii)

From (i) and (ii), $A = - 1$ and $B = 1$.