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#