How do you determine if the improper integral converges or diverges #int (1/(3x)-6) dx# from negative infinity to 0?

Answer 1

#int_-oo^0(1/(3x)-6)dxrArr# diverges to #-oo#

To evaluate a two-sided improper integral, split it into two integrals and express each as a limit like so:

#int_-oo^0(1/(3x)-6)dx#
#=lim_(s->-oo)int_s^-1(1/(3x)-6)dx+lim_(t->0)int_-1^t(1/(3x)-6)dx#

Now compute the integrals and evaluate the limits:

#rArrlim_(s->-oo)[lnabs(3x)/3-6x]_s^-1#
#rArrlim_(s->-oo)[(ln3/3+6)-(lnabs(3s)/3-6s)]#
#rArr[(ln3/3+6)-(oo+oo)]->-oo#
Therefore the integral diverges as it approaches #-oo#.

Technically, we could stop here. If one integral diverges, the whole expression will diverge. But let's evaluate the second limit just for good measure:

#rArrlim_(t->0)[lnabs(3x)/3-6x]_-1^t#
#rArrlim_(t->0)[(lnabs(3t)/3-6t)-(ln3/3+6)]#
#rArr[(-oo-0)-(ln3/3+6)]->-oo#

And again we see the integral also diverges as it approaches 0.

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Answer 2

To determine if the improper integral converges or diverges, we need to evaluate the limit of the integral as the upper limit approaches 0 from the left (since the lower limit is negative infinity). We can express the integral as follows:

∫(from -∞ to 0) (1/(3x) - 6) dx

To integrate (1/(3x) - 6), we need to split it into two separate integrals:

∫(from -∞ to 0) (1/(3x)) dx - ∫(from -∞ to 0) 6 dx

The first integral, ∫(1/(3x)) dx, is an improper integral that diverges. This can be shown by evaluating the limit of the integral as x approaches 0 from the left:

lim(x→0-) ∫(from -∞ to x) (1/(3x)) dx = lim(x→0-) [ln|3x|] (from -∞ to x) = lim(x→0-) [ln|3x| - ln|3(-∞)|] = ∞

The second integral, ∫(6) dx, is a finite integral:

∫(from -∞ to 0) 6 dx = [6x] (from -∞ to 0) = 6(0) - 6(-∞) = 0 - (-∞) = ∞

Since one part of the integral diverges (∫(1/(3x)) dx) and the other part converges (∫(6) dx), the overall integral diverges.

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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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