# How do you find the limit #(t+5-2/t-1/t^3)/(3t+12-1/t^2)# as #x->oo#?

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To find the limit of the given expression as x approaches infinity, we can simplify it by dividing every term by the highest power of x in the denominator.

The expression can be rewritten as: [(t + 5 - 2/t) / (t - 1)] / [(3t + 12 - 1/t^2)]

Now, let's simplify each term separately:

As t approaches infinity, the term 2/t becomes negligible, and we can ignore it.

In the numerator, (t + 5) / (t - 1) simplifies to 1.

In the denominator, (3t + 12) / t^2 simplifies to 3/t.

Therefore, the simplified expression is 1 / (3/t).

To simplify further, we can multiply the numerator and denominator by t:

1 / (3/t) = t / 3

As t approaches infinity, the limit of t/3 is infinity.

Hence, the limit of the given expression as t approaches infinity is infinity.

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