Problem 14 - Calculus Early Transcendentals

Question

Use Theorem 3.11 to evaluate the following limits. $$\lim _{\theta \rightarrow 0} \frac{\sec \theta-1}{\theta}$$

Step-by-Step Solution

Answer

Question: Evaluate the limit: $\lim _{\theta \rightarrow 0} \frac{\sec \theta-1}{\theta}$ Answer: 0

Step 1: Rewrite the limit as per Theorem 3.11

Let us consider \(f(\theta) = \sec \theta\). In this case, we can write the given limit as follows: $$\lim _{\theta \rightarrow 0} \frac{f(\theta)-f(0)}{\theta - 0}$$ where \(c = 0\) and \(f(0) = \sec 0 = 1\). Therefore, the limit is: $$\lim _{\theta \rightarrow 0} \frac{\sec \theta-1}{\theta} = \lim_{\theta \to 0} \frac{f(\theta) - f(0)}{\theta - 0} = f'(0)$$ Now, we need to find the derivative \(f'(\theta)\).

Step 2: Find the derivative of f

Let's find the derivative of the function \(f(\theta) = \sec \theta\) using basic differentiation rules. The derivative of the secant function is \(f'(\theta) = \sec \theta \tan \theta\).

Step 3: Evaluate f'(0)

Now that we have the derivative \(f'(\theta) = \sec \theta \tan \theta\), let's evaluate it at \(\theta = 0\). We have: $$f'(0) = \sec{0} \tan{0} = 1 \cdot 0 = 0$$

Step 4: Substitute f'(0) in the limit

Finally, substitute the value of \(f'(0)\) in the limit expression as per Theorem 3.11: $$\lim _{\theta \rightarrow 0} \frac{\sec \theta-1}{\theta} = f'(0) = 0$$ So, the value of this limit is 0.

Key Concepts

Derivative of secant function

The derivative of the secant function, \(\sec \theta\), is a fundamental concept in calculus and is important when we deal with trigonometric limits and derivatives. The secant function is defined as the reciprocal of the cosine function, i.e., \(\sec \theta = \frac{1}{\cos \theta}\). When differentiating \(\sec \theta\), we utilize basic differentiation rules. The quotient rule or the chain rule are often used ways to find it. Specifically, the derivative of the secant function is:

• \(f'(\theta) = \sec \theta \tan \theta\)

This result is arrived by expressing \(\sec \theta\) in terms of \(\frac{1}{\cos \theta}\) and differentiating accordingly with the chain rule. Whenever the derivative is positive, the secant function is increasing, and if it’s negative, the function decreases. Knowing how to differentiate trigonometric functions like \(\sec \theta\) is crucial for solving calculus problems where these functions appear in limits.

Limits approaching zero

Limits are essential in calculus for understanding the behavior of functions as they approach specific points. Specifically, limits approaching zero are fundamental when evaluating derivatives or determining the continuity of a function. The limit \(\lim_{\theta \rightarrow 0} \) essentially asks what value a function is tending towards as \(\theta\) gets very close to zero.In our specific problem, the limit \(\lim_{\theta \rightarrow 0} \frac{\sec \theta - 1}{\theta}\) involves the secant function minus 1 over \(\theta\). This expression is significant as it is structured similarly to the definition of a derivative:

• \(\lim_{\theta \rightarrow c} \frac{f(\theta) - f(c)}{\theta - c}\)

These types of limits are often evaluated by finding the derivative, even when the original problem doesn’t explicitly mention differentiation. This connection between limits and derivatives underscores the intrinsic relationship between these concepts in calculus.

Differentiation rules

Differentiation rules are pivotal in calculus, providing the tools to determine the rate at which functions change. The problem in this exercise uses basic differentiation rules to find the derivative of the secant function. Crucial differentiation rules include:

• Power Rule: If \(f(x) = x^n\), then \(f'(x) = nx^{n-1}\).
• Constant Rule: The derivative of a constant is 0.
• Sum Rule: The derivative of a sum is the sum of the derivatives.
• Product Rule: If \(f(x) = u(x)v(x)\), then \(f'(x) = u'(x)v(x) + u(x)v'(x)\).
• Quotient Rule: If \(f(x) = \frac{u(x)}{v(x)}\), then \(f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{v(x)^2}\).
• Chain Rule: If \(f(x) = g(h(x))\), then \(f'(x) = g'(h(x))h'(x)\).

For the secant function, \(\sec \theta = \frac{1}{\cos \theta}\), applying the differentiation rules precisely gives us \(f'(\theta) = \sec \theta \tan \theta\) as shown in the solution. Understanding these rules is important to compute derivatives efficiently, especially in solving more complex calculus problems.