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Chapter 2: Problem 1

Evaluate the limits that exist. $$\lim _{x \rightarrow 0} \frac{\sin 3 x}{x}$$

Short Answer

Expert verified
The value of \(\lim _{x \rightarrow 0} \frac{\sin 3 x}{x}\) is 3.

Step by step solution

01

Write down the given expression

The given expression is: \(\lim _{x \rightarrow 0} \frac{\sin 3 x}{x}\)
02

Apply the limit property

According to a known property, \(\lim_{x \rightarrow 0} \frac{{\sin{x}}}{x} = 1\). However, the expression inside the sine function in the given problem is \(3x\) rather than \(x\). By a trick of multiplying and dividing by the same number (3 in this case), we can apply the limit property as in: \(\lim _{x \rightarrow 0} \frac{\sin 3 x}{3x} * 3\)
03

Calculate the limit

Applying limit properties to the expression, we get: \(\lim_{x \rightarrow 0} \frac{\sin 3x}{3x} * 3 = 1 * 3 = 3\)

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Key Concepts

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

Trigonometric Limits
Trigonometric limits are a key concept in calculus, especially when dealing with trigonometric functions' behavior as they approach certain points.
Understanding the behavior of functions like sine and cosine is crucial when evaluating limits where the variable approaches a specific value.
  • For instance, a commonly used trigonometric limit is \(\lim_{x \to 0} \frac{\sin{x}}{x} = 1\). This is a foundational limit used in various calculus problems, including the one discussed here.
  • When you see expressions like \(\frac{\sin{kx}}{x}\), where \(k\) is a constant, you often utilize this property to simplify the evaluation of the limit by ensuring the argument of the sine function and the denominator match.
Applying these concepts allows simplification and the right calculations. So, recognizing and applying these standard limits is an essential skill when tackling trigonometric problems in calculus.
Limit Properties
Limit properties are rules and formulas that facilitate the evaluation of limits, making complex problems simpler and manageable.
These properties allow us to manipulate the limits of functions and make conclusions based on known behaviors.
  • For example, if you know that \(\lim_{x \to a} f(x) = L\) and \(\lim_{x \to a} g(x) = M\), then the limit properties can tell you that \(\lim_{x \to a} (f(x) + g(x)) = L + M\) and \(\lim_{x \to a} (f(x) \cdot g(x)) = L \cdot M\).
  • Another crucial property is the limit of a product. In this mode, you can split a complex expression into manageable pieces, as seen when the original problem used \(\lim_{x \to 0} \frac{\sin{3x}}{3x} \cdot 3\). This is a clever manipulation where the expression is split, and limits are calculated for each part separately.
This way, we see how limit properties not only apply to simple arithmetic operations but can also make solving more complex trigonometric limits much easier.
Calculus Techniques
Calculus provides various techniques for evaluating limits, including algebraic manipulation, substitution, and the use of known limit properties.
These methods build on each other to break down complex problems into simpler forms.
  • One such technique is substitution or a trick of multiplying and dividing, which was used in this problem where multiplying and dividing by 3 helps apply a known trigonometric limit effectively.
  • This arises from an understanding of trigonometric identities and the behavior of functions at points close to zero.
By employing such calculus techniques, you approach tricky limits with confidence, using logical steps to simplify and solve them. By mastering these strategies, complex calculus problems become much more manageable and less intimidating.

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Most popular questions from this chapter

True or false? Justify your answers. $$\begin{aligned} &\text { If } \lim _{x \rightarrow c}[f(x)+g(x)] \text { exist but } \lim _{x \rightarrow c} f(x) \text { does not exist, then }\\\ &\lim _{x \rightarrow c} g(x) \text { does not exist. } \end{aligned}$$

After estimating the limit using the prescribed values of \(x,\) validate or improve your estimate by using a graphing utility. Estimate $$\lim _{x \rightarrow 0} \frac{1-\cos x}{x} \quad \text { (radian measure) }$$

Evaluate the limits that exist. $$\lim _{x \rightarrow \pi / 2} \frac{\cos x}{x-\frac{1}{2} \pi}$$

Show that \(\lim _{x \rightarrow 0} x \sin (1 / x)=0 .\) HDN: Use the pinching theorem.

Evaluate the limits that exist. $$\lim _{x \rightarrow \pi / 4} \frac{\sin x}{x}$$
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