Problem 81
Question
Consider the limit \(\lim _{x \rightarrow 0^{+}}(-x \ln x)\) (a) Describe the type of indeterminate form that is obtained by direct substitution. (b) Evaluate the limit. (c) Use a graphing utility to verify the result of part (b). FOR FURTHER INFORMATION For a geometric approach to this exercise, see the article "A Geometric Proof of \(\lim _{l \rightarrow 0^{+}}(-d \ln d)=0\) " by John H. Mathews in the College Mathematics Journal. To view this article, go to the website www.matharticles.com.
Step-by-Step Solution
Verified Answer
The indeterminate form is \(0 \cdot \infty\), the limit is 0, and the graph of \(-x \ln x\) as \(x\) approaches \(0^{+}\) confirms this result.
1Step 1: Determine the Indeterminate Form
When \(x\) substitutes to \(0^{+}\), the function \(-x \ln x\) becomes \(-0( \ln 0) = -0(\infty)\), which is an indeterminate form, specifically \(0 \cdot \infty\).
2Step 2: Apply L'Hopital's Rule and Evaluate the Limit
In order to apply L'Hopital's Rule, rewrite the limit in the form \(\lim _{x \rightarrow 0^{+}}\frac{\ln x}{-1/x}\). This now has the indeterminate form \(\frac{-\infty}{\infty}\), for which L'Hopital's Rule can be used. Take the derivative of the numerator and denominator to get \(\lim _{x \rightarrow 0^{+}}\frac{1/x}{1/x^{2}} = \lim _{x \rightarrow 0^{+}}x = 0\).
3Step 3: Use a Graphing Utility
Input \(-x\ln x\) into a graphing utility. As \(x\) approaches \(0^{+}\), you should observe the value of the function approaching zero, confirming the result obtained analytically.
Key Concepts
L'Hopital's RuleIndeterminate FormLimitsGraphing Utilities
L'Hopital's Rule
L'Hopital's Rule is a tool in calculus for solving limits that result in an indeterminate form. It simplifies the process by allowing us to differentiate the numerator and the denominator separately. This can only be applied when the initial form of the limit is indeterminate, such as \( \frac{0}{0} \) or \( \frac{\pm\infty}{\pm\infty} \).
The rule states that \[ \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} \] provided the limit on the right-hand side exists. It’s important to apply L'Hopital’s Rule only when necessary and after verifying the conditions, as it relies on differentiability of the functions involved. A typical mistake is using it without confirming the indeterminate form.
Understanding this concept is key for successful application in various limit problems.
The rule states that \[ \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} \] provided the limit on the right-hand side exists. It’s important to apply L'Hopital’s Rule only when necessary and after verifying the conditions, as it relies on differentiability of the functions involved. A typical mistake is using it without confirming the indeterminate form.
Understanding this concept is key for successful application in various limit problems.
Indeterminate Form
Indeterminate forms are certain expressions where the direct substitution of a value does not lead to a clear answer. Common examples include \(0 \cdot \infty\), \(\frac{0}{0}\), and \(\frac{\infty}{\infty}\).
When we tried to substitute \(x = 0^+\) into \(-x \ln x\), we uncovered the form \(-0(\ln 0)\), equating to \(0 \cdot \infty\). This is considered indeterminate because it doesn’t immediately suggest a limit value.
Because these forms give no clear answer, we use techniques such as algebraic manipulation or L'Hopital's Rule to further analyze. Recognizing indeterminate forms is crucial because they identify situations where basic arithmetic does not apply and further analysis is required to find the limit.
When we tried to substitute \(x = 0^+\) into \(-x \ln x\), we uncovered the form \(-0(\ln 0)\), equating to \(0 \cdot \infty\). This is considered indeterminate because it doesn’t immediately suggest a limit value.
Because these forms give no clear answer, we use techniques such as algebraic manipulation or L'Hopital's Rule to further analyze. Recognizing indeterminate forms is crucial because they identify situations where basic arithmetic does not apply and further analysis is required to find the limit.
Limits
Limits are fundamental in calculus, used to understand the behavior of functions as they approach specific points. When evaluating limits, we want to identify what value a function approaches as the input gets closer to a particular point.
In our exercise, we looked at \(\lim _{x \to 0^+}(-x \ln x)\). We noticed the behavior of the function is unclear when using direct substitution due to the indeterminate form \(0 \cdot \infty\).
L'Hopital's Rule helped us transform this complex expression into a simple calculation. After applying the rule and re-evaluating the limit, we found the answer to be \(0\). Understanding limits enables us to comprehend continuity, derivatives, and integrals—key concepts in calculus.
In our exercise, we looked at \(\lim _{x \to 0^+}(-x \ln x)\). We noticed the behavior of the function is unclear when using direct substitution due to the indeterminate form \(0 \cdot \infty\).
L'Hopital's Rule helped us transform this complex expression into a simple calculation. After applying the rule and re-evaluating the limit, we found the answer to be \(0\). Understanding limits enables us to comprehend continuity, derivatives, and integrals—key concepts in calculus.
Graphing Utilities
Graphing utilities are indispensable tools for visualizing functions and supporting analytical calculations. They confirm results derived algebraically by providing graphical insights.
In our exercise, after finding the limit analytically, using a graphing utility like Desmos or a graphing calculator helps verify the result. By inputting the function \(-x \ln x\) and observing its behavior as \(x \to 0^+\), we can see the function’s value approaches \(0\), which aligns with our analytical finding.
These tools are user-friendly and provide immediate visual representation, helping reinforce or challenge our understanding of the problem. Utilizing graphing utilities can also reveal additional insights, such as asymptotic behavior or function intersections, which might not be initially evident through calculation alone.
In our exercise, after finding the limit analytically, using a graphing utility like Desmos or a graphing calculator helps verify the result. By inputting the function \(-x \ln x\) and observing its behavior as \(x \to 0^+\), we can see the function’s value approaches \(0\), which aligns with our analytical finding.
These tools are user-friendly and provide immediate visual representation, helping reinforce or challenge our understanding of the problem. Utilizing graphing utilities can also reveal additional insights, such as asymptotic behavior or function intersections, which might not be initially evident through calculation alone.
Other exercises in this chapter
Problem 80
Use a graphing utility to graph \(f(x)=\frac{x^{k}-1}{k}\) for \(k=1,0.1\), and 0.01 . Then evaluate the limit \(\lim _{k \rightarrow 0^{+}} \frac{x^{k}-1}{k}\)
View solution Problem 81
Laplace Transforms Let \(f(t)\) be a function defined for all positive values of \(t\). The Laplace Transform of \(f(t)\) is defined by \(F(s)=\int_{0}^{\infty}
View solution Problem 82
Laplace Transforms Let \(f(t)\) be a function defined for all positive values of \(t\). The Laplace Transform of \(f(t)\) is defined by \(F(s)=\int_{0}^{\infty}
View solution Problem 82
Prove that if \(f(x) \geq 0, \lim _{x \rightarrow a} f(x)=0,\) and \(\lim _{x \rightarrow a} g(x)=\infty,\) then \(\lim _{x \rightarrow a} f(x)^{g(x)}=0\).
View solution