Consider the difference:
#abs( (2n^2+1)/(7n^2+n+5) -2/7) = abs ( (14n^2+7-14n^2-2n-10)/(49n^2+7n+35))#
#abs( (2n^2+1)/(7n^2+n+5) -2/7) =(2n+3)/(49n^2+7n+35)#
#abs( (2n^2+1)/(7n^2+n+5) -2/7) =(2/n+3/n^2)/(49+7/n+35/n^2)#
As the denominator is always greater than #49#:
#abs( (2n^2+1)/(7n^2+n+5) -2/7) <=1/49(2/n+3/n^2)#
and as: #n^2 >=n#:
#abs( (2n^2+1)/(7n^2+n+5) -2/7) <=5/49*1/n#
Given any #epsilon >0# choose then #N_epsilon > 5/(49epsilon)#, then, for #n >N_epsilon#
#abs( (2n^2+1)/(7n^2+n+5) -2/7) <=5/49*1/n < 5/49*1/N_epsilon < 5/49 *(49epsilon)/5 = epsilon#
In other words:
#AA epsilon > 0, EE N_epsilon: n> N_epsilon => abs( (2n^2+1)/(7n^2+n+5) -2/7) < epsilon#
which proves the limit.
