Question

Find the limit. Use l'Hospital's Rule if appropriate. If there is a more elementary method, consider using it. limx--->0 (In(x)) /2

Answer 1

The limit cannot be evaluated using l'Hospital's rule. As x approaches 0, the natural logarithm function In(x) approaches negative infinity, so the limit is -∞.

Step-by-step explanation:

To find the limit:

limx--->0 (In(x)) / 2

We can use l'Hospital's rule to evaluate the limit. To do this, we differentiate the numerator and denominator separately.

Let's start by differentiating the numerator:

d/dx(In(x)) = 1/x

Now, let's differentiate the denominator:

d/dx(2) = 0

Next, we take the limit of the quotient of the derivatives:

limx--->0 (1/x) / 0

This expression is of the form 1/0, which is undefined. Therefore, we cannot evaluate the limit using l'Hospital's rule.

However, we can use a more elementary method to find the limit. As x approaches 0, the natural logarithm function In(x) approaches negative infinity. So, the limit is:

limx--->0 (In(x)) / 2 = -∞ / 2 = -∞.