Problem 6 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Evaluate the following limits. Explain your reasoning. (a) \(\lim _{x \rightarrow \infty} \sin x\) (b) \(\lim _{x \rightarrow \infty} \frac{\sin x}{x}\)

Step-by-Step Solution

Answer

The limit of \(\sin x\) as x approaches infinity does not exist and the limit of \(\frac{\sin x}{x}\) as x approaches infinity is 0.

Step 1: Determine the limit of sin(x) as x approaches infinity

The function \(\sin x\) doesn't have a limit as x approaches infinity. The reason is that the value of \(\sin x\) oscillates between -1 and 1 for all real numbers, so it doesn’t settle down to any one value as x grows larger and larger. Therefore, \(\lim _{x \rightarrow \infty} \sin x\) doesn't exist.

Step 2: Determine the limit of \(\frac{\sin x}{x}\) as x approaches infinity

As x approaches infinity, the function \(\frac{\sin x}{x}\) approaches 0. When x gets very large, the denominator grows significantly faster than the numerator (which stays within the range [-1,1]), causing the ratio to approach 0. Thus, \(\lim _{x \rightarrow \infty} \frac{\sin x}{x} = 0\).

Key Concepts

Limits

In calculus, limits are fundamental concepts used to understand the behavior of functions as they approach a specific point or infinity. When we say "the limit of a function as x approaches a certain value," we mean where the function is heading towards as x gets closer to that value. Limits are essential for defining derivatives, integrals, and for evaluating functions at points where they may not be straightforward.For instance, assessing limits can help us understand the behavior of a function such as \( \lim_{x \to \infty} f(x) \), which tells us what happens to the function values as x becomes very large. Calculating limits can involve simple observation of function behavior or more complex techniques like using L'Hôpital's rule or series expansions. In the context of sinusoidal functions approaching infinity, limits become particularly interesting due to their oscillatory nature.

Sinusoidal Functions

Sinusoidal functions like \( \sin x \) and \( \cos x \) are periodic and oscillate between -1 and 1. This oscillation means that these functions continuously go up and down in a wave-like pattern. Such behavior makes them fascinating yet tricky to analyze as x approaches infinity.With sinusoidal functions, their periodic nature implies they never settle down at a particular value over time, which is key when analyzing limits. For example, when looking at \( \lim_{x \to \infty} \sin x \), the result is undefined because the oscillating pattern prevents \( \sin x \) from approaching any single value. It perpetually bounces between -1 and 1, thereby not having a finite limit as x grows.This concept also shows why sinusoidal functions are unique in calculus. Despite not having a limit at infinity, they allow us to explore other fascinating phenomena, such as analyzing their behavior in ratios as in the limit of \( \frac{\sin x}{x} \).

Infinity

Infinity is a concept rather than a number, representing something that is unending or boundless. In calculus, approaching infinity is an essential idea when examining limits, derivatives, and integrals.When we evaluate statements like \( \lim_{x \to \infty} f(x) \), we are interested in understanding how a function behaves as its input grows without bound. For many functions, approaching infinity might lead to the function stabilizing at a certain value or growing without limit itself.In the context of calculus problems involving sinusoidal functions or fractions, understanding infinity helps in discerning why certain limits exist or do not. For example, in \( \lim_{x \to \infty} \frac{\sin x}{x} \), although \( \sin x \) never settles as x goes to infinity due to its periodic nature, the growing denominator x makes the fraction

• smaller and smaller
• tending towards zero.

This showcases how infinity often acts as a balancing factor that can simplify or define the behavior of intricate functions like sinusoidal ratios.