Evaluate the following limits :
`Lim_(x to oo) (cos x)/x `

Answer

To evaluate the limit \( \lim_{x \to \infty} \frac{\cos x}{x} \), we can follow these steps:

### Step 1: Analyze the function
The function \( \cos x \) oscillates between -1 and 1 for all values of \( x \). Therefore, we can write:
\[ -1 \leq \cos x \leq 1 \]

### Step 2: Divide the inequality by \( x \)
Since \( x \) is approaching infinity and is always positive for large \( x \), we can divide the entire inequality by \( x \):
\[ -\frac{1}{x} \leq \frac{\cos x}{x} \leq \frac{1}{x} \]

### Step 3: Evaluate the limits of the bounding functions
Now, we will evaluate the limits of the bounding functions as \( x \) approaches infinity:
1. For the lower bound:
\[ \lim_{x \to \infty} -\frac{1}{x} = 0 \]
2. For the upper bound:
\[ \lim_{x \to \infty} \frac{1}{x} = 0 \]

### Step 4: Apply the Squeeze Theorem
Since \( \frac{\cos x}{x} \) is squeezed between \( -\frac{1}{x} \) and \( \frac{1}{x} \), and both of these limits approach 0, we can conclude by the Squeeze Theorem:
\[ \lim_{x \to \infty} \frac{\cos x}{x} = 0 \]

### Final Answer
Thus, the limit is:
\[ \lim_{x \to \infty} \frac{\cos x}{x} = 0 \]

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