Finding Limits


In Exercises 9–24, find the limit or explain why it does not exist.




lim x →π sin (x/2 + sin x)

Verified step by step guidance

1. Step 1: Understand the problem. We need to find the limit of the function sin(x/2 + sin(x)) as x approaches π.
2. Step 2: Substitute x = π into the expression inside the sine function. This gives us π/2 + sin(π). Since sin(π) = 0, the expression simplifies to π/2.
3. Step 3: Evaluate the sine function at π/2. We know that sin(π/2) = 1.
4. Step 4: Since the function sin(x/2 + sin(x)) is continuous around x = π, the limit as x approaches π is simply the value of the function at x = π.
5. Step 5: Conclude that the limit of sin(x/2 + sin(x)) as x approaches π is 1, based on the continuity and substitution.

Key Concepts

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near specific points, which is crucial for defining derivatives and integrals. Limits can exist or not exist, and determining their existence often involves evaluating the function at points close to the limit.

Continuity

Continuity refers to a property of functions where they do not have any abrupt changes, jumps, or holes at a given point. A function is continuous at a point if the limit as the input approaches that point equals the function's value at that point. Understanding continuity is essential for evaluating limits, as discontinuities can lead to limits that do not exist.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are periodic functions that relate angles to ratios of sides in right triangles. In the context of limits, these functions can exhibit specific behaviors as their arguments approach certain values, which can affect the limit's existence. Familiarity with the properties and values of trigonometric functions is crucial for evaluating limits involving them.