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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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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