Please help me with Lim excercise and explain it as well? Really appreciate!

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Answer 1 · 2018-04-12T18:46:12

Assuming $nrarroo$ , we have $a = 1$ .

$lim_(nrarroo)[sqrt(2n^2+n)-asqrt(2n^2-n)]$

Observe that if $a = -1$ , then the limit is $oo$ (not real).

So we know that $a != -1$

Rewrite. Multiply by $(sqrt(2n^2+n)+asqrt(2n^2-n))/(sqrt(2n^2+n)+asqrt(2n^2-n))$ to get:

$sqrt(2n^2+n)-a^2sqrt(2n^2-n) = ((2n^2+n)-a^2(2n^2-n))/(sqrt(2n^2+n)+a^2sqrt(2n^2-n))$

$= (2n^2(1-a^2)-n(1+a^2))/ (n(sqrt(2+1/n)+a^2sqrt(2-1/n))$

$= (2n(1-a^2)-(1+a^2))/ (sqrt(2+1/n)+a^2sqrt(2-1/n)$

Now if $1-a^2 != 0$ , then the limit is $oo$ (not real).

So $1-a^2 = 0$ .

We already ruled out $-1$ , so we must have $a = 1$