Question

Use Theorem 3. 10 to evaluate the following limits. $$\lim _{x \rightarrow 0} \frac{\sin a x}{b x}, \text { where } a \text { and } b \text { are constants with } b \neq 0$$

Short answer

Answer: The limit of the expression \(\frac{\sin ax}{bx}\) as \(x\) approaches \(0\) is \(\frac{a}{b}\).

Step-by-step solution

Write down the given limit

Given the limit: $$\lim_{x \rightarrow 0} \frac{\sin ax}{bx}$$

Use a substitution

Let \(y = ax\). Then as \(x\) approaches \(0\), \(y\) also approaches \(0\). Applying the substitution, the limit now becomes: $$\lim_{y \rightarrow 0} \frac{\sin y}{b\frac{y}{a}}$$

Simplify the expression

Re-write the limit expression as: $$\lim_{y \rightarrow 0} \frac{a}{b} \cdot \frac{\sin y}{y}$$

Apply Theorem 3.10

Now, apply Theorem 3.10, which states that \(\lim_{y \rightarrow 0} \frac{\sin y}{y} = 1\): $$\frac{a}{b} \cdot \lim_{y \rightarrow 0} \frac{\sin y}{y} = \frac{a}{b} \cdot 1$$

Compute the final result

The result of the limit is: $$\lim_{x \rightarrow 0} \frac{\sin ax}{bx} = \frac{a}{b}$$

Key concepts

Theorem 3.10

When studying limits, especially those involving trigonometric functions, Theorem 3.10 stands out as a crucial principle. It succinctly states that \( \lim_{y \rightarrow 0} \frac{\sin y}{y} = 1 \). This theorem is important because it applies to a special case where a sine function is being divided by its own angle in radians when that angle approaches zero. This outcome isn't immediately obvious through algebraic manipulation, but a geometric or an analytic justification can confirm its validity.

This theorem is frequently applied when dealing with trigonometric limits and it's often used in tandem with other limit properties to find the limit of more complex expressions. The elegance of this approach lies in its simplicity and its far-reaching applicability to a wide variety of problems in calculus.

Sine Function Limits

Considering the sine function when dealing with limits is pivotal in calculus. The sine function exhibits a unique behavior as the input approaches zero. This is quantified by the previously mentioned Theorem 3.10, which can be thought of as a foundational block for understanding sine function limits. Additionally, limits of the sine function as the argument approaches infinity or some other value may also be of interest, but often require different tools to evaluate.

Knowing how to manipulate and simplify expressions involving sine functions to make use of limit theorems ensures students can tackle a wider range of problems. This skill becomes particularly important in higher-level mathematics courses that deal with series and more intricate functions.

Limit Substitution Method

When confronted with complex limits, the substitution method often simplifies the process. In substitution, a new variable is introduced, replacing a part of the original function to make it easier to evaluate the limit. As seen in the exercise, the substitution \( y = ax \) is used where the limit is then evaluated at \( y \) approaching zero, which is an equivalent scenario to \( x \) approaching zero when \( a \) is a constant.

Substitution is not always straightforward – it requires a keen eye to identify what replacement will lead to a solvable limit. The main advantage of this method is that it can turn a problematic limit into one that is more manageable and directly applicable to known limit theorems or properties.

Trigonometric Limits

Trigonometric functions, which include sine, cosine, and tangent functions, often appear in limit problems. They can prove tricky due to their oscillatory behavior. Trigonometric limits are a subset of limits that specifically deal with these types of functions as the variable approaches a particular value. The process often involves understanding and applying specific limit theorems, like Theorem 3.10, or using L'Hôpital's rule when dealing with forms that involve division by zero or infinity.

As is demonstrated in this exercise, familiarity with trigonometric identities and properties can also be instrumental in evaluating these limits. For instance, the limit of the sine function divided by its angle as it approaches zero is one such property that is frequently used. By mastering these concepts, students can accurately and confidently solve calculus problems that involve evaluating trigonometric limits.