Prove the following limit, where the Squeeze Theorem is used: lim, 0(xtan-1(x)) = 0
Prove the following limit, where the Squeeze Theorem is used: lim, 0(xtan-1(x)) = 0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![**Topic: Limit Calculations Using the Squeeze Theorem**
**Objective:** Prove the following limit using the Squeeze Theorem.
\[ \lim_{x \to 0} (x \tan^{-1}(x)) = 0 \]
**Explanation:**
In this exercise, we aim to demonstrate that as \( x \) approaches zero, the expression \( x \tan^{-1}(x) \) approaches zero, employing the Squeeze Theorem.
The Squeeze Theorem is a mathematical principle used to find the limit of a function by "squeezing" it between two other functions that have the same limit at a specific point. To apply this theorem effectively, we need to identify two functions, \( g(x) \) and \( h(x) \), such that \( g(x) \leq f(x) \leq h(x) \), and both \( \lim_{x \to a} g(x) \) and \( \lim_{x \to a} h(x) \) equal the same value, which will also be the limit of \( f(x) \) at that point.
For this particular problem, consider the properties of the inverse tangent function and its behavior around zero. The next steps would involve calculating and finding appropriate bounding functions and carrying out the necessary algebraic manipulation to complete the proof.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbaa9c753-6b68-41b4-8837-61aa8aa0a19b%2F8405a8c8-33a7-4e0b-9dfb-92d24ed7f1f8%2Focke2ei_processed.png&w=3840&q=75)
Transcribed Image Text:**Topic: Limit Calculations Using the Squeeze Theorem**
**Objective:** Prove the following limit using the Squeeze Theorem.
\[ \lim_{x \to 0} (x \tan^{-1}(x)) = 0 \]
**Explanation:**
In this exercise, we aim to demonstrate that as \( x \) approaches zero, the expression \( x \tan^{-1}(x) \) approaches zero, employing the Squeeze Theorem.
The Squeeze Theorem is a mathematical principle used to find the limit of a function by "squeezing" it between two other functions that have the same limit at a specific point. To apply this theorem effectively, we need to identify two functions, \( g(x) \) and \( h(x) \), such that \( g(x) \leq f(x) \leq h(x) \), and both \( \lim_{x \to a} g(x) \) and \( \lim_{x \to a} h(x) \) equal the same value, which will also be the limit of \( f(x) \) at that point.
For this particular problem, consider the properties of the inverse tangent function and its behavior around zero. The next steps would involve calculating and finding appropriate bounding functions and carrying out the necessary algebraic manipulation to complete the proof.
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