Limits and Continuity

Evaluating Limits Involving Trigonometric Functions

Limits involving trigonometric functions are a fundamental topic in calculus, especially when analyzing the behavior of functions as the variable approaches a specific value. A common technique is to use trigonometric identities and standard limit results to simplify and evaluate these limits.

• Key Trigonometric Identity: \( \cos x = 1 - 2 \sin^2 \left( \frac{x}{2} \right) \)

• Standard Limit: \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \)

Example: Evaluating a Trigonometric Limit

Consider the limit:

\[ \lim_{x \to 0} \frac{-2 \sin^2 \left( \frac{x}{2} \right)}{x} \]

Using the identity for \( \cos x \), we can rewrite and simplify the expression:

1. Express \( \sin^2 \left( \frac{x}{2} \right) \) in terms of \( x \):

   • \( \sin^2 \left( \frac{x}{2} \right) = \left( \sin \left( \frac{x}{2} \right) \right)^2 \)

2. Rewrite the limit:

   • \( \lim_{x \to 0} \frac{-2 \sin^2 \left( \frac{x}{2} \right)}{x} = \lim_{x \to 0} -2 \cdot \frac{\sin \left( \frac{x}{2} \right)}{x} \cdot \sin \left( \frac{x}{2} \right) \)

3. Express \( \frac{\sin \left( \frac{x}{2} \right)}{x} \) as \( \frac{\sin \left( \frac{x}{2} \right)}{\frac{x}{2}} \cdot \frac{1}{2} \):

   • \( \frac{\sin \left( \frac{x}{2} \right)}{x} = \frac{\sin \left( \frac{x}{2} \right)}{\frac{x}{2}} \cdot \frac{1}{2} \)

4. Apply the standard limit result:

   • \( \lim_{x \to 0} \frac{\sin \left( \frac{x}{2} \right)}{\frac{x}{2}} = 1 \)

5. Combine the results:

   • \( \lim_{x \to 0} -2 \cdot 1 \cdot \frac{1}{2} = -1 \)

However, the detailed calculation on the board shows the following steps:

• \( \lim_{x \to 0} -2 \sin^2 \left( \frac{x}{2} \right) / x = \lim_{x \to 0} -2 \cdot \frac{\sin \left( \frac{x}{2} \right)}{x} \cdot \sin \left( \frac{x}{2} \right) \)

• \( = \lim_{x \to 0} -2 \cdot \left( \frac{\sin \left( \frac{x}{2} \right)}{\frac{x}{2}} \right)^2 \cdot \left( \frac{x}{2} \right)^2 / x \)

• \( = -2 \cdot 1^2 \cdot \frac{1}{4} = -\frac{1}{2} \)

Final Answer:

\[ \lim_{x \to 0} \frac{-2 \sin^2 \left( \frac{x}{2} \right)}{x} = -\frac{1}{2} \]

Additional info:

• This example demonstrates the use of trigonometric identities and the standard limit \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \) to evaluate more complex limits.

• Such techniques are essential for solving limits that appear indeterminate at first glance.