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How do you determine the limit of # ((x^2 +2)/(x^3-1))# as x approaches 0?

Answer 1

I found #-2#

I would try by approaching zero from the left and the right where you get, in both cases, that your function get as near as possible to the value #-2#.

Your function is also continuous in #x=0# giving you:

#lim_(x->0)f(x)=f(x_0)#

i.e., the limit at zero is equal to the function evaluated at zero or: #f(0)=((0^2+2)/(0^3-1))=-2#

I would say that the limit exists and is equal to #-2#.

Graphically you can see this as well:

graph{(x^2+2)/(x^3-1) [-10, 10, -5, 5]}

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Answer 2

Luke Alvarez

To determine the limit of ((x^2 + 2)/(x^3 - 1)) as x approaches 0, we can substitute 0 into the expression and simplify. By substituting 0 for x, we get (0^2 + 2)/(0^3 - 1), which simplifies to 2/(-1). Therefore, the limit of ((x^2 + 2)/(x^3 - 1)) as x approaches 0 is -2.

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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.