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Chapter 10: Problem 3

\(\lim _{x \rightarrow 0} \frac{\sin 9 x}{\sin 7 x}\)

Short Answer

Expert verified
The limit is \(\frac{9}{7}\).

Step by step solution

01

Recall the Sine Limit Property

Recall that \(\frac{\text{sin}(x)}{x} \rightarrow 1 \text{ as } x \rightarrow 0\). This will help simplify the expression.
02

Rewrite the Expression

Rewrite the given limit in terms of the property: \(\frac{\text{sin}(9x)}{\text{sin}(7x)} = \frac{\text{sin}(9x)/(9x)}{\text{sin}(7x)/(7x)} \times \frac{9x}{7x}\).
03

Apply the Limit

Apply the limit using the property from Step 1. Recognize that, as \( x \rightarrow 0 \), \( \frac{\text{sin}(9x)}{9x} \rightarrow 1\) and \( \frac{\text{sin}(7x)}{7x} \rightarrow 1\). This leaves us with: \(\frac{1}{1} \times \frac{9x}{7x} = \frac{9}{7} \).
04

Confirm the Result

After applying the limits, confirm the result: \( \frac{9}{7} \).

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Key Concepts

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

Sine Function
The sine function, often written as \(\text{sin}(x)\), is a fundamental function in trigonometry. It's essential to understand how it behaves, especially as the input approaches certain values.
  • The sine function is periodic with a period of 2\( \pi \).
  • When the input value, x, approaches 0, \(\text{sin}(x)\) behaves very predictably.
  • This predictable behavior is crucial in calculus as it forms the basis for various limit properties.
For small values of x, the sine function approximates the value of x itself. This property is often used to simplify calculations involving limits.
Limit Properties
Limits are a core concept in calculus. A limit evaluates what a function's output approaches as the input approaches a particular value.
  • One vital property is the limit of the sine function as the input approaches zero.
  • If you encounter \(\frac{\text{sin}(x)}{x}\), it approaches 1 as x approaches 0. This is because \(\text{sin}(x)\) starts to behave like x for very small values.
Understanding and utilizing limit properties makes solving calculus problems easier. These properties allow breaking down complex problems into understandable parts.
Trigonometric Limits
Trigonometric limits specifically deal with limits involving trigonometric functions like sine, cosine, and tangent.
  • These limits are crucial because trigonometric functions often arise in various calculus problems.
  • A key limit to remember is \( \lim_{x \to 0} \frac{\text{sin}(x)}{x} = 1 \). This serves as a foundation for solving more complex trigonometric limits.
Using the trigonometric limit property, complex trigonometric limits can be simplified, making them easier to solve.
L'H么pital's Rule
L'H么pital's Rule is a powerful tool for evaluating limits, especially when direct substitution in limit problems results in indeterminate forms, like \( \frac{0}{0} \) or \( \frac{\text{infinity}}{\text{infinity}} \).
  • The rule states that if \(\frac{f(x)}{g(x)}\) approaches \( \frac{0}{0} \) or \( \frac{\text{infinity}}{\text{infinity}} \), then \( \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \).
  • This means you can differentiate the numerator and the denominator separately and then take the limit of the resulting function.
It's essential to use this rule correctly as it simplifies complex problems by providing a way to break down the indeterminate forms.

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Most popular questions from this chapter

\(\int \frac{\csc ^{4} x}{\cot ^{2} x} d x\)

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