Evaluating the Limit of (x^n ln(1-x)) as (x) Approaches Zero
In this article, we will explore the limit of the function (x^n ln(1-x)) as (x) approaches zero. This involves understanding the behavior of the function when (x 0) and applying various mathematical techniques such as l'H?pital's rule. This comprehensive guide will help you through the process with detailed steps and explanations.
Understanding the Function
Consider the function (f(x) x^n ln(1-x)). When (n geq 0), the function is well-defined and continuous at (x 0). This means we can substitute (x 0) directly into the function to find the limit, which is (0). However, when (n
When (n
For the case where (n
[lim_{x to 0} frac{ln(1-x)}{x^m}]
where (m -n). This is a useful substitution that transforms the original function into a more manageable form. Let's proceed step-by-step to find the limit.
Applying L'H?pital's Rule
When we substitute (x 0) into (frac{ln(1-x)}{x^m}), we encounter a (frac{0}{0}) indeterminate form, ideal for applying L'H?pital's rule. L'H?pital's rule states that if the limit of a quotient of two functions is of the form (frac{0}{0}) or (frac{infty}{infty}), the limit of the quotient is equal to the limit of the quotient of their derivatives.
Applying L'H?pital's rule, we get:
[lim_{x to 0} frac{ln(1-x)}{x^m} lim_{x to 0} frac{-frac{1}{1-x}}{mx^{m-1}}]
Now, we can simplify and evaluate the limit:
[ lim_{x to 0} frac{-1}{mx^{m-1}}]
Case Analysis
We now consider different cases for (m).
Case 1: (m 1)
When (m 1), the expression simplifies to:
[lim_{x to 0} frac{-1}{1-x} 0]
Thus, the limit is (0).
Case 2: (m > 1)
When (m > 1), the expression becomes:
[-frac{1}{mx^{m-1}}]
As (x) approaches (0), (m-1 > 0), and thus (mx^{m-1}) approaches (0). Therefore, the limit is:
[-infty]
So, the limit does not exist for (m > 1).
Case 3: (0
When (0
[-frac{1}{mx^{m-1}}]
Here, (m-1
[-frac{1}{infty} 0]
Hence, the limit from the right-hand side is (0).
Conclusion
Summarizing:
When (n geq 0), the limit of (x^n ln(1-x)) as (x) approaches (0) is (0). When (0 When (m > 1), the limit does not exist.Understanding these cases can help in evaluating similar limits involving logarithmic and polynomial functions. If you have any further questions or need a more detailed explanation, feel free to ask!