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Chapter 2: Problem 31

Find the limit, if it exists. $$\lim _{h \rightarrow 0} \frac{\sin x \sin h}{h}$$

Short Answer

Expert verified
The limit is \( \text{sin}(x) \).

Step by step solution

01

Identify the limit to be evaluated

Given the limit \[ \text{lim}_{h \to 0} \frac{\text{sin}(x) \text{sin}(h)}{h} \], we need to find its value as \( h \) approaches 0.
02

Use trigonometric properties

We know from trigonometric identities that \( \text{sin}(x) \) is a constant with respect to \( h \). Thus, it can be factored out of the limit.
03

Factor out the constant

Factor out \( \text{sin}(x) \) from the limit expression: \[ \text{lim}_{h \to 0} \frac{\text{sin}(x) \text{sin}(h)}{h} = \text{sin}(x) \text{lim}_{h \to 0} \frac{\text{sin}(h)}{h} \]
04

Evaluate the remaining limit

We use the standard limit \[ \text{lim}_{h \to 0} \frac{\text{sin}(h)}{h} = 1 \].
05

Multiply the results

Substitute the value of the standard limit into our expression: \[ \text{sin}(x) \times 1 = \text{sin}(x) \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sin(x) Limit
Understanding the limit involving \text{sin}(x) and how it behaves as a variable approaches a certain value is crucial in calculus. In this exercise, we are given the limit expression \(\text{lim}_{h \to 0} \frac{\text{sin}(x) \text{sin}(h)}{h} \). To solve this, we need to break it down into manageable parts. Because \text{sin}(x) does not depend on \( h \), it can be treated as a constant. This allows us to separate \text{sin}(x) from the limit operation. By factoring out \text{sin}(x) from the expression, we can focus on evaluating \(\text{lim}_{h \to 0} \frac{\text{sin}(h)}{h} \), which is a well-known limit in trigonometry.
Trigonometric Identities
Trigonometric identities are tools that allow us to simplify and solve complex trigonometric expressions. In this problem, knowing that \text{sin}(x) is a constant with respect to \( h \) helps us simplify the given limit. Remember, when dealing with limits, constants can be factored out. This is why we rewrite the original expression as follows: \(\text{lim}_{h \to 0} \frac{\text{sin}(x) \text{sin}(h)}{h} = \text{sin}(x) \text{lim}_{h \to 0} \frac{\text{sin}(h)}{h} \). Factoring out the constant allows us to utilize the trigonometric limit identity.
Limit Evaluation
Evaluating the limit of trigonometric functions often requires knowledge of standard limits. One such fundamental limit is \(\text{lim}_{h \to 0} \frac{\text{sin}(h)}{h} = 1 \). This identity is pivotal and frequently used in calculus to solve problems involving trigonometric functions. Substituting this known limit back into our expression \(\text{sin}(x) \text{lim}_{h \to 0} \frac{\text{sin}(h)}{h} \), we get \(\text{sin}(x) \times 1 \), which simplifies to \(\text{sin}(x) \). This step-by-step breakdown highlights the importance of recognizing and applying fundamental limits to simplify and solve calculus problems efficiently.

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