Problem 44 - Calculus: Early Transcendentals

Question

Determine the following limits. $$ \lim _{\theta \rightarrow 0^{-}} \frac{\sin \theta}{\cos ^{2} \theta-1} $$

Step-by-Step Solution

Answer

Question: Determine the value of the following trigonometric limit, if possible: $$ \lim _{\theta \rightarrow 0^{-}} \frac{\sin \theta}{\cos ^{2} \theta-1} $$ Answer: The limit is undefined.

Step 1: Use trigonometric identity

Notice that the denominator is \(\cos^2\theta - 1\). We can use the trigonometric identity \(\cos^2\theta - \sin^2\theta = 1\) to transform this expression: $$ \cos^2\theta - 1 = -(1 - \cos^2\theta) = -\sin^2\theta $$ Now, the limit becomes: $$ \lim _{\theta \rightarrow 0^{-}} \frac{\sin \theta}{-\sin^2 \theta} $$

Step 2: Simplify expression

We can now simplify this expression by dividing \(\sin\theta\) in the numerator by \(\sin^2\theta\) in the denominator: $$ \lim _{\theta \rightarrow 0^{-}} \frac{\sin\theta}{-\sin^2\theta} = -\lim _{\theta \rightarrow 0^{-}} \frac{1}{\sin\theta} $$

Step 3: Evaluate limit at 0

Now we can directly evaluate the limit as \(\theta\rightarrow 0^{-}\): $$ -\lim _{\theta \rightarrow 0^{-}} \frac{1}{\sin\theta} = -\frac{1}{\sin(0^{-})} = -\frac{1}{0} $$ This limit is undefined, because the sine function approaches 0 when its argument approaches 0. Therefore, the initial limit does not have a finite value. Answer: The limit is undefined.

Key Concepts

Trigonometric Limits

Trigonometric limits are a fundamental concept in calculus. They are used to evaluate the behavior of trigonometric functions as they approach a particular point or value. Understanding trigonometric limits helps us to understand the precision and accuracy of trigonometric functions at specific values.

A very common technique is the substitution of trigonometric identities. For the limit \( \lim_{\theta \to 0^{-}} \) given in the problem, we had to substitute and simplify terms to evaluate it accurately. This limit is particularly interesting because it requires careful handling of division by zero, a common challenge with trigonometric limits.

When evaluating these types of limits, one often needs to use L'Hôpital's rule when encountering indeterminate forms like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). In this exercise, the denominator approached zero, which is a hint that careful attention and additional techniques might be needed to determine if the limit can be defined.

Trigonometric Identities

Trigonometric identities play a crucial role in simplifying expressions involving trigonometric functions. They are equations that are true for every value of the occurring variables where both sides of the identity are defined.

One of the primary identities used in many calculus problems is the Pythagorean identity: \(\cos^2\theta + \sin^2\theta = 1\). This identity can be rearranged to show other equations such as \(\cos^2\theta = 1 - \sin^2\theta\), which is key to solving the original limit problem by transforming the denominator.

In calculus, these identities often help us rewrite limits into forms that are more easily evaluated. By using identities, complex expressions become simpler, allowing us to apply basic limit laws.“ Then, when we have simplified an expression using identities, we can more easily take the next steps in evaluating a limit.

Sine and Cosine Functions

The sine and cosine functions are fundamental in trigonometry. They oscillate in a wave-like pattern and have properties that make them essential in many calculus problems.

Sine functions, denoted as \(\sin(\theta)\), depend on the angle \(\theta\) and have a range from -1 to 1. Understanding the behavior of sine around zero is crucial in problem-solving, particularly for limits that involve the function approaching zero.

Similarly, cosine functions, represented as \(\cos(\theta)\), have the same range and are often used alongside sine in trigonometric identities. The fact that \(\cos(0) = 1\) is a key property, often used in limit problems.

• Sine and cosine are periodic with a period of \(2\pi\), meaning they repeat every \(2\pi\) units.
• Their continuity and differentiability are advantageous when calculating limits.
• They are often involved in common trigonometric identities that simplify calculus operations.

Mastering sine and cosine properties ensures better comprehension of calculus problems, especially when handling limits and identities together.