Problem 14 - Calculus for Biology and Medicine

Question

Evaluate the trigonometric limits. $$ \lim _{x \rightarrow 0} \frac{1-\cos ^{2} x}{x^{2}} $$

Step-by-Step Solution

Answer

The limit evaluates to 1.

Step 1: Recognize the Expression

The given limit is \( \lim _{x \rightarrow 0} \frac{1 - \cos^2 x}{x^2} \). Recognize that \(1 - \cos^2 x\) is an identity related to sine: \(1 - \cos^2 x = \sin^2 x\).

Step 2: Rewrite Using Identity

Rewrite the given limit expression using the identity \(1 - \cos^2 x = \sin^2 x\), resulting in \(\lim _{x \rightarrow 0} \frac{\sin^2 x}{x^2}\).

Step 3: Apply Limit Laws

The limit \(\lim _{x \rightarrow 0} \frac{\sin^2 x}{x^2}\) can be rewritten as \(\left(\lim_{x\rightarrow 0} \frac{\sin x}{x}\right)^2\). The well-known limit \(\lim_{x\rightarrow 0} \frac{\sin x}{x} = 1\) can be applied.

Step 4: Compute the Final Limit

Since \(\lim_{x\rightarrow 0} \frac{\sin x}{x} = 1\), substitute into the expression: \(\left(\lim_{x\rightarrow 0} \frac{\sin x}{x}\right)^2 = 1^2 = 1\). Thus, the value of the limit is 1.

Key Concepts

Limit Laws

Limit Laws are essential tools in calculus. They help evaluate the behavior of functions as inputs get arbitrarily close to a certain point. In the context of trigonometric limits, these laws are particularly helpful because they simplify otherwise complex problems.
When using limit laws, remember a few key points:

• Limits of individual terms can be combined, i.e., the limit of a sum is the sum of the limits.
• For products, the limit of a product is the product of the limits.
• The same applies for quotients, provided the limit of the denominator is not zero.

In the exercise example given, the limit \[\lim_{x \rightarrow 0} \frac{\sin^2 x}{x^2}\]can be separated into \[\left(\lim_{x\rightarrow 0} \frac{\sin x}{x} \right)^2\]through application of the quotient and power rule of limits. This breakdown allows us to apply known limits, such as \[\lim_{x\rightarrow 0} \frac{\sin x}{x} = 1\]. By doing so, calculating the limit becomes direct and manageable.

Trigonometric Identities

Trigonometric Identities are equations involving trigonometric functions that hold for all values of the variables involved. They are essential in manipulating and simplifying trigonometric expressions, especially in calculus.
One classic identity is \[1 - \cos^2 x = \sin^2 x\]which plays a crucial role in simplifying limits like the one in the exercise. By recognizing and substituting this identity, the problem becomes much more manageable.
The power of these identities lies in their ability to transform complex trigonometric expressions into simpler forms, which are often easier to handle. Other key identities include:

• \(\sin^2 x + \cos^2 x = 1\)
• \(\tan^2 x + 1 = \sec^2 x\)
• \(1 + \cot^2 x = \csc^2 x\)

These identities facilitate a wide range of mathematical problems beyond limits, including integration and differentiation, making them indispensable tools for solving trigonometric equations.

Sine and Cosine Functions

Sine and Cosine are fundamental trigonometric functions that describe the ratios of sides in a right-angled triangle. Understanding their properties is crucial in evaluating trigonometric limits.
The Sin function \(\sin x\) is periodic, with values ranging between -1 and 1. Its curve starts at the origin and completes a full cycle every \(2\pi\). Similarly, the Cosine function \(\cos x\) has the same range and periodicity, but starts at 1.
When working with limits involving these functions, it's helpful to remember:

• Both \(\sin x\) and \(\cos x\) are continuously differentiable functions.
• Near zero, \(\sin x\) approximates to \(x\) and both approach 0 as \(x\) approaches 0.
• The identity \(\lim_{x \rightarrow 0} \frac{\sin x}{x} = 1\) is especially helpful.

By understanding these properties, evaluating limits that seem complex at first glance becomes more straightforward. The sine and cosine functions are widely used in various fields including physics and engineering, emphasizing their importance in mathematical applications.