It means that x approaches zero from the right (positive values close to zero).phillyolly said:Homework Statement
lim(x→0^+)(lnx/x)
The Attempt at a Solution
First, what does x->0+ mean? Is it positive infinity?
Look at the graph of y = ln(x). The domain is {x | x > 0}. As x gets closer to zero, y gets more and more negative.phillyolly said:Why lim(x→0^+) (lnx)=-∞?
What you have above is mostly incorrect. 1/0 is not a number, so you can't use it in calculations. [tex]\lim_{x \to 0^+} \frac{ln(x)}{x} = \lim_{x \to 0^+} \frac{1}{x}ln(x) [/tex]phillyolly said:Why the answer is
= -∞ * (1/0+)
= -∞* ∞
= -∞?
Read the fine print in L'Hopital's Rule. There are certain conditions that must be satisfied before you can use it.phillyolly said:Why cannot I use l'Hospital Rule?
The limit of a function as x approaches 0+ is the value that the function approaches from the right side of the x-axis. In other words, it is the value that the function gets closer and closer to as x gets closer and closer to 0 from the positive side.
L'Hospital's Rule is a mathematical rule that can be used to help evaluate the limit of a function in cases where the limit may be indeterminate, such as when both the numerator and denominator approach 0. It involves taking the derivative of the numerator and denominator separately and then evaluating the limit again.
L'Hospital's Rule should be used when the limit of a function is indeterminate, which occurs when both the numerator and denominator approach 0, infinity, or negative infinity. It can also be used when the limit is a 0/0 or infinity/infinity form.
Yes, there are a few limitations to using l'Hospital's Rule. It can only be used when the limit is indeterminate, and it can only be used to evaluate the limit of a rational function (a function with a polynomial in the numerator and denominator). Additionally, it should only be used as a last resort after trying other methods to evaluate the limit.
Yes, l'Hospital's Rule can be used for one-sided limits. When evaluating a one-sided limit, you would take the derivative of the numerator and denominator separately and then evaluate the limit again using the appropriate one-sided approach (i.e. approaching from the left or right side of the x-axis).