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Magazine Chapter 6: Problem 35
Find each limit. (a) \(\lim _{x \rightarrow 1^{-}} \sin ^{-1} x\) (b) \(\lim _{x \rightarrow-1^{+}} \sin ^{-1} x\)
Short Answer
Expert verified
(a) \(\frac{\pi}{2}\); (b) \(-\frac{\pi}{2}\).
Step by step solution
01
Understanding Inverse Sine Function
The inverse sine function, denoted as \( ext{arcsin}(x)\) or \( ext{sin}^{-1}(x)\), is defined for \(-1 \leq x \leq 1\), and it yields angles in the range \(-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}\).
02
Analyze the Left-Hand Limit at x = 1
For part (a), we need to find \(\lim_{x \to 1^-} \sin^{-1}(x)\). As \(x\) approaches 1 from the left, the values of \(\sin^{-1}(x)\) approach \(\frac{\pi}{2}\), since \(\sin^{-1}(1) = \frac{\pi}{2}\).
03
Evaluate the Left-Hand Limit
Evaluate the limit directly: \(\lim_{x \to 1^-} \sin^{-1}(x) = \frac{\pi}{2}\). This is because the inverse sine function approaches \(\frac{\pi}{2}\) smoothly as \(x\) approaches 1 from the left.
04
Analyze the Right-Hand Limit at x = -1
For part (b), we need to find \(\lim_{x \to -1^+} \sin^{-1}(x)\). As \(x\) approaches \(-1\) from the right, the values of \(\sin^{-1}(x)\) approach \(-\frac{\pi}{2}\), since \(\sin^{-1}(-1) = -\frac{\pi}{2}\).
05
Evaluate the Right-Hand Limit
Evaluate the limit directly: \(\lim_{x \to -1^+} \sin^{-1}(x) = -\frac{\pi}{2}\). This limit evaluates to \(-\frac{\pi}{2}\) as the inverse sine function approaches this value smoothly from the right.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Limits
Limits are a fundamental concept in calculus. They help us understand the behavior of a function as it approaches a particular point. We often write the limit of a function \( f(x) \) as \( x \) approaches a number \( c \) in the form \( \lim_{x \to c} f(x) \).
- If a function approaches the same value from both the left and the right as \( x \) approaches \( c \), it has a limit at that point.
- Limits are used to study continuity, derivatives, and integrals.
- They can also help us understand functions' behaviors, such as whether they reach a maximum, minimum, or asymptote.
Left-Hand Limit
The left-hand limit describes the behavior of a function as the variable approaches a specific point from the left. Mathematically, it is written as \( \lim_{x \to c^-} f(x) \). This notation indicates that \( x \) is getting closer to \( c \) from values less than \( c \).
- Calculating the left-hand limit involves examining the function's values as \( x \) nears \( c \) from below.
- In our exercise, we evaluated \( \lim_{x \to 1^-} \sin^{-1}(x) = \frac{\pi}{2} \). As \( x \) approaches 1 from values less than 1, \( \sin^{-1}(x) \) tends smoothly towards \( \frac{\pi}{2} \).
Right-Hand Limit
The right-hand limit examines how a function behaves as the variable approaches a specified point from the right. It is denoted as \( \lim_{x \to c^+} f(x) \), implying that \( x \) approaches \( c \) from values greater than \( c \).
- To find the right-hand limit, analyze the function's behavior as \( x \) nears \( c \) from above.
- In the provided exercise, we saw that \( \lim_{x \to -1^+} \sin^{-1}(x) = -\frac{\pi}{2} \). As \( x \) approaches -1 from values greater than -1, the inverse sine function smoothly tends towards \(-\frac{\pi}{2} \).
Arcsin Function
The arcsin function, or inverse sine function, is denoted by \( \sin^{-1}(x) \) or \( \text{arcsin}(x) \). It is the inverse of the sine function and is defined for inputs between \(-1\) and \(1\).
- The range of arcsin is \([-\frac{\pi}{2}, \frac{\pi}{2}]\), meaning it outputs angles (in radians).
- For example, \( \sin^{-1}(0) = 0 \), \( \sin^{-1}(1) = \frac{\pi}{2} \), and \( \sin^{-1}(-1) = -\frac{\pi}{2} \).
- The arcsin function is crucial in trigonometry, helping us to find angles when the sine value is known.
