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

Answer 1

# lim_(xrarroo)(x+5-2/x-1/x^3)/(3x+12-1/x^2) = 1/3 #

I assume that you mean #lim_(xrarroo)(x+5-2/x-1/x^3)/(3x+12-1/x^2) # and that the expression should not contain the variable #t#!!
Now, As #x->oo# then #1/x->0#
So, it would be better if we could replace #x# with #1/x#
# lim_(xrarroo)(x+5-2/x-1/x^3)/(3x+12-1/x^2) = lim_(xrarroo)(1/x(x+5-2/x-1/x^3))/(1/x(3x+12-1/x^2)) #
# :. lim_(xrarroo)(x+5-2/x-1/x^3)/(3x+12-1/x^2) = lim_(xrarroo)(1+5/x-2/x^2-1/x^4)/(3+12/x-1/x^3) #
# :. lim_(xrarroo)(x+5-2/x-1/x^3)/(3x+12-1/x^2) = (1+0-0-0)/(3+0-0)=1/3 #
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Answer 2

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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Answer from HIX Tutor

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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