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

Answer

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

### Step 1: Understand the behavior of \( \sin x \)
The function \( \sin x \) oscillates between -1 and 1 for all values of \( x \). Therefore, we can say:
\[ -1 \leq \sin x \leq 1 \]

### Step 2: Divide by \( x \)
Now, we can divide the entire inequality by \( x \) (keeping in mind that \( x \) is positive as \( x \to \infty \)):
\[ -\frac{1}{x} \leq \frac{\sin x}{x} \leq \frac{1}{x} \]

### Step 3: Analyze the limits of the bounding functions
Next, we will evaluate the limits of the bounding functions as \( x \) approaches infinity:
\[ \lim_{x \to \infty} -\frac{1}{x} = 0 \]
\[ \lim_{x \to \infty} \frac{1}{x} = 0 \]

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

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

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