Final answer:
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 = -∞.