How do you evaluate the limit #(x^3+2)/(x+1)# as x approaches #oo#?

Answer 1

#lim_(x->oo) (x^3+2)/(x+1) =+oo#

You can separate the numerator in two parts, one of which can be divided by (x+1). For instance add and subtract #1# and you get:
#(x^3+2)/(x+1) = (x^3 +1 +2-1)/(x+1) =(x^3+1)/(x+1) +1/(x+1)#
As #bar x = -1# is a root of #x^3+1#, then #(x+1)# must be a factor, and in fact:
#(x^3 +1) = (x^2-x+1)(x+1)#

so that:

#(x^3+2)/(x+1) =((x^2-x+1)(x+1))/(x+1) +1/(x+1) = (x^2-x+1) +1/(x+1)#

Now go for the limit: the first addendum tends to infinity, the second addendum tends to zero.

#lim_(x->oo) (x^3+2)/(x+1) = lim_(x->oo)[(x^2-x+1) +1/(x+1)] =+oo#
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Answer 2

To evaluate the limit of (x^3+2)/(x+1) as x approaches infinity, we can use the concept of limits at infinity.

As x approaches infinity, the highest power term in the numerator (x^3) and the highest power term in the denominator (x) dominate the expression.

Dividing each term in the numerator and denominator by x, we get (x^3/x + 2/x)/(x/x + 1/x).

Simplifying this expression, we have (x^2 + 2/x)/(1 + 1/x).

As x approaches infinity, the term 2/x becomes negligible compared to x^2, and 1/x becomes negligible compared to 1.

Therefore, the expression simplifies to x^2/1, which is equal to x^2.

Hence, the limit of (x^3+2)/(x+1) as x 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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