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

Evaluating the integral shows that it diverges

Therefore this integral diverges

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To determine the convergence or divergence of the improper integral (\int_{-\infty}^0 \frac{1}{3x - 6} , dx), we need to analyze its behavior as (x) approaches negative infinity.

First, rewrite the integral as: [\int_{-\infty}^0 \frac{1}{3x - 6} , dx = \lim_{a \to -\infty} \int_a^0 \frac{1}{3x - 6} , dx]

Now, find the antiderivative of (\frac{1}{3x - 6}): [F(x) = \frac{1}{3} \ln|3x - 6|]

Apply the Fundamental Theorem of Calculus: [\lim_{a \to -\infty} F(0) - F(a)]

[= \lim_{a \to -\infty} \left(\frac{1}{3} \ln|3(0) - 6|\right) - \left(\frac{1}{3} \ln|3a - 6|\right)]

[= \lim_{a \to -\infty} \frac{1}{3} (\ln|-6| - \ln|3a - 6|)]

[= \lim_{a \to -\infty} \frac{1}{3} \ln\left(\frac{-6}{3a - 6}\right)]

[= \lim_{a \to -\infty} \ln\left(\left(\frac{-2}{a - 2}\right)^{\frac{1}{3}}\right)]

Since the exponent approaches zero as (a) approaches negative infinity, the expression inside the natural logarithm approaches 1.

Thus, the integral converges.

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To determine whether the improper integral ( \int_{-\infty}^0 \frac{1}{3x - 6} , dx ) converges or diverges, we first need to analyze its behavior as ( x ) approaches negative infinity and as it approaches zero.

As ( x ) approaches negative infinity, the function ( \frac{1}{3x - 6} ) approaches zero. Therefore, we need to check the behavior of the function near the singularity at ( x = 2 ).

Near ( x = 2 ), the function ( \frac{1}{3x - 6} ) behaves like ( \frac{1}{3(x-2)} ). As ( x ) approaches 2 from the left (i.e., as ( x ) approaches 2 from negative infinity), the function approaches negative infinity.

Therefore, the integral is improper at ( x = 2 ), and we need to check the convergence around this point. Since the function approaches negative infinity near ( x = 2 ), the integral diverges.

Hence, the improper integral ( \int_{-\infty}^0 \frac{1}{3x - 6} , dx ) diverges from negative infinity to zero.

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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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