Limits and Infinity


Find the limits in Exercises 37–46.


sin x
lim ------------- ( If you have a grapher, try graphing
x→∞ |x| the function for ―5 ≤ x ≤ 5 ) .

Verified step by step guidance

1. Identify the function for which you need to find the limit: \( \frac{\sin x}{|x|} \).
2. Understand that as \( x \to \infty \), the absolute value \( |x| \) behaves like \( x \) because \( x \) is positive.
3. Recognize that the sine function, \( \sin x \), oscillates between -1 and 1 for all real numbers \( x \).
4. Consider the behavior of the fraction \( \frac{\sin x}{x} \) as \( x \to \infty \). Since \( \sin x \) is bounded and \( x \) grows without bound, the fraction approaches zero.
5. Conclude that the limit of \( \frac{\sin x}{|x|} \) as \( x \to \infty \) is 0, because the numerator is bounded while the denominator increases indefinitely.

Key Concepts

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They help in understanding the behavior of functions at specific points, including points of discontinuity or infinity. In this context, evaluating the limit as x approaches infinity allows us to analyze the long-term behavior of the function sin(x)/|x|.

Behavior of Functions at Infinity

When analyzing limits as x approaches infinity, we assess how a function behaves as its input grows without bound. This often involves determining whether the function approaches a finite value, diverges to infinity, or oscillates. For the function sin(x)/|x|, understanding its behavior as x becomes very large is crucial for finding the limit.

Trigonometric Functions

Trigonometric functions, such as sine, exhibit periodic behavior, oscillating between fixed values. The function sin(x) oscillates between -1 and 1, which is important when considering its limit in conjunction with |x|. This periodic nature influences the overall limit of the function sin(x)/|x| as x approaches infinity, as the oscillation will be divided by an increasingly large denominator.