`a_n = (10n^2+3n+7)/(2n^2-6)` Determine the convergence or divergence of the sequence with the given n'th term. If the sequence converges, find its limit.

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`lim_(n to infty)(10n^2+3n+7)/(2n^2-6)=`

Divide both numerator and the denominator by `n^2.`

`lim_(n to infty)((10n^2)/n^2+(3n)/n^2+7/n^2)/((2n^2)/n^2-6/n^2)=lim_(n to infty)(10+3/n+7/n^2)/(2-6/n^2)=`

Since `lim_(n to infty) alpha/n^beta=0,` `forallalpha in RR,` `forall beta>0` we have


As we can see the sequence converges to 5.

The image below shows first 150 terms of the sequence. We can see they are asymptotically approaching the red line whose equation is `y=5.` 

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