find the limit using l'hospital's rule

lim x-->negative infinity (x ln(1-1/x)

x ln(1 - 1/x) = ln(1 - 1/x) / (1/x)

so the limit is the same as
[1/(1 - 1/x) * (1/x^2)] / (-1/x^2) = -1/(1 - 1/x)
in the limit as x→∞, that is just -1

To find the limit using L'Hospital's rule, follow these steps:

Step 1: Rewrite the expression in a form that allows the application of L'Hospital's rule. Here, we have the limit:

lim x→-∞ (x ln(1 - 1/x))

Since we have an indeterminate form of (-∞ * 0), we can rewrite it as:

lim x→-∞ [ln(1 - 1/x) / (1/x)]

Step 2: Differentiate the numerator and denominator separately.

Taking the derivative of the numerator:

d/dx ln(1 - 1/x) = (1/(1 - 1/x)) * (d/dx (1 - 1/x))
= (1/(1 - 1/x)) * (0 + 1/(x^2))
= 1/(1 - 1/x) * 1/(x^2)
= x^2/(x^2 - x)

Differentiating the denominator:

d/dx (1/x) = -1/x^2

Step 3: Simplify the expression.

lim x→-∞ [x^2/(x^2 - x) / (-1/x^2)]
= lim x→-∞ [-x^4/(x^2 - x)]

Step 4: Evaluate the limit.

Now, when x approaches negative infinity, the term (-x^4) dominates both the numerator and denominator. As a result, the limit becomes:

lim x→-∞ [-x^4/(x^2 - x)] = ∞

Therefore, the limit of the expression as x approaches negative infinity is ∞.

To find the limit as x approaches negative infinity of x ln(1 - 1/x) using L'Hospital's rule, we can apply the rule of differentiation to the numerator and denominator separately.

Let's begin by differentiating the numerator. The derivative of x with respect to x is simply 1. For the derivative of ln(1 - 1/x), we can use the chain rule.

The derivative of ln(u) with respect to u is 1/u. Therefore, the derivative of ln(1 - 1/x) with respect to x is (-1/(1 - 1/x)) * (-1/x^2).

Now let's differentiate the denominator, which is just 1.

Taking the limit as x approaches negative infinity, we have:

lim x --> -∞ (1/((-1/(1 - 1/x)) * (-1/x^2)))

Simplifying this expression further, we get:

lim x --> -∞ ((x^2) / (1 - 1/x))

Now, we can substitute negative infinity into the expression:

lim x --> -∞ ((-∞^2) / (1 - 1/(-∞)))

Since (∞^2) is positive infinity and (1 - 1/(-∞)) is 1, the limit becomes:

lim x --> -∞ (∞ / 1)

Finally, we have:

lim x --> -∞ (∞)

Therefore, the limit as x approaches negative infinity of x ln(1 - 1/x) using L'Hospital's rule is infinity.