Question

If $$\displaystyle \lim_{n \rightarrow \infty} \displaystyle \left ( an - \frac{1 + n^2}{1 + n} \right ) = b$$, where a and b are finite numbers, then

A
a=1
B
a=0
C
b=1
D
b=1

Solution

The correct options are B $$b = 1$$ C $$a = 1$$$$limit \displaystyle = \lim_{n \rightarrow \infty} \frac{an ( 1 + n) - (1 + n^2)}{1 + n}$$$$=\displaystyle \lim_{n \rightarrow \infty} \displaystyle \frac{(a-1) n^2 + an - 1}{n + 1}$$$$= \infty$$ if $$a - 1 \neq 0$$If $$a- 1 = 0$$, limit $$=\displaystyle \lim_{n \rightarrow \infty } \displaystyle \frac{an - 1}{n + 1} = a = b$$$$\therefore a = b = 1$$Maths

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