How do you show whether the improper integral #int (79 x^2/(9 + x^6)) dx# converges or diverges from negative infinity to infinity?
To determine the convergence or divergence of the improper integral ( \int_{\infty}^{\infty} \frac{79x^2}{9 + x^6} , dx ), we'll use the comparison test.

Consider the behavior of ( \frac{79x^2}{9 + x^6} ) as ( x ) approaches ( \pm \infty ).
As ( x ) approaches ( \pm \infty ), the term ( x^6 ) dominates, and the function behaves like ( \frac{79x^2}{x^6} = \frac{79}{x^4} ).

Since the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator, the function approaches zero as ( x ) approaches ( \pm \infty ).

We can compare ( \frac{79x^2}{9 + x^6} ) with ( \frac{79}{x^4} ) for large ( x ) values.
As ( x ) approaches ( \pm \infty ): [ \frac{79x^2}{9 + x^6} < \frac{79}{x^4} ]

The integral ( \int_{\infty}^{\infty} \frac{79}{x^4} , dx ) can be evaluated as a pseries with ( p = 4 ), and it converges since ( p > 1 ).

By the Comparison Test, since ( \frac{79x^2}{9 + x^6} < \frac{79}{x^4} ) and ( \int_{\infty}^{\infty} \frac{79}{x^4} , dx ) converges, ( \int_{\infty}^{\infty} \frac{79x^2}{9 + x^6} , dx ) converges as well.
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I would integrate by trigonometric substitution, then check that the limit exists.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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