Does the function converge or diverge #1/((x-1)(x^2+1)) #on the bound [2,infinity]?

Answer 1

Assuming poles are meant, #x=+-1#

The nomenclature is a bit unclear - the given function doesn't really "converge" or "diverge" in the way it is specified - these are more properties of sequences or series. Functions can be specified as series, a manner in which they can converge or diverge for various values, but that doesn't seem to apply here.

I'm going to assume that what is meant is to find the #x# values for which the function blows up to infinity. These occur when the denominator is equal to zero. In this case the denominator has handily already been factored for us, and has three roots with two distinct values, #+-1#.
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Answer 2

To determine if the function ( \frac{1}{(x-1)(x^2+1)} ) converges or diverges on the interval ([2, \infty)), we need to check for convergence or divergence of the function as ( x ) approaches infinity. We can do this by analyzing the behavior of the function at large values of ( x ).

As ( x ) approaches infinity, the terms ( (x-1) ) and ( (x^2+1) ) become dominant. Since ( x^2 ) grows faster than ( x ), the term ( (x^2 + 1) ) dominates the denominator.

Therefore, as ( x ) approaches infinity, the function behaves like ( \frac{1}{x^2} ), which converges. Hence, the function ( \frac{1}{(x-1)(x^2+1)} ) converges on the interval ([2, \infty)).

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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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