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Chapter 3: Problem 54

Evaluate the following limits or state that they do not exist. $$\lim _{x \rightarrow \infty} \frac{\cos x}{x}$$

Short Answer

Expert verified
Answer: The limit of the function \(\frac{\cos x}{x}\) as x approaches infinity is 0.

Step by step solution

01

Identify the Limit

We are given the limit: $$\lim_{x \rightarrow \infty} \frac{\cos x}{x}$$ To solve this, we need to analyze the behavior of the ratio as x approaches infinity.
02

Analyze the Cosine Function

Remember that the cosine function is periodic, with a period of \(2\pi\) (360 degrees). This means that \(\cos x\) oscillates between -1 and 1 for all real numbers x: $$-1 \leq \cos x \leq 1$$
03

Analyze the Denominator

As x approaches infinity, the denominator also increases without bound.
04

Determine Behavior of the Ratio

Since the numerator (the cosine function) is trapped within the range of -1 and 1. Whereas the denominator increases indefinitely as x approaches infinity, the overall effect will be that the fraction \(\frac{\cos x}{x}\) is driven towards zero. Mathematically, we can express this as: $$\lim _{x \rightarrow \infty} \frac{\cos x}{x} = \lim_{x \rightarrow \infty} \frac{-1}{x} \leq \lim_{x \rightarrow \infty} \frac{\cos x}{x} \leq \lim_{x \rightarrow \infty} \frac{1}{x}$$ Since \(\lim_{x \rightarrow \infty} \frac{-1}{x} = \lim_{x \rightarrow \infty} \frac{1}{x} = 0\), by the Squeeze Theorem, we have: $$\lim _{x \rightarrow \infty} \frac{\cos x}{x} = 0$$

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