Find the asymptotes of the following curves :
`y = 2x - arc cos . 1/x `

Answer

To find the asymptotes of the curve given by the equation \( y = 2x - \arccos\left(\frac{1}{x}\right) \), we will follow these steps:

### Step 1: Identify the function
We have the function:
\[ y = f(x) = 2x - \arccos\left(\frac{1}{x}\right) \]

### Step 2: Analyze the behavior as \( x \to \infty \)
To find the asymptotes, we need to analyze the limit of \( f(x) \) as \( x \) approaches infinity:
\[ \lim_{x \to \infty} f(x) = \lim_{x \to \infty} \left( 2x - \arccos\left(\frac{1}{x}\right) \right) \]

### Step 3: Evaluate \( \arccos\left(\frac{1}{x}\right) \)
As \( x \to \infty \), \( \frac{1}{x} \to 0 \). Therefore, we can find:
\[ \arccos\left(\frac{1}{x}\right) \to \arccos(0) = \frac{\pi}{2} \]

### Step 4: Substitute back into the limit
Now substituting this back into our limit:
\[ \lim_{x \to \infty} f(x) = \lim_{x \to \infty} \left( 2x - \frac{\pi}{2} \right) \]
As \( x \to \infty \), \( 2x \) dominates, and thus:
\[ \lim_{x \to \infty} f(x) = \infty \]

### Step 5: Find the asymptotic behavior
To find the asymptotic behavior, we need to express \( f(x) \) in the form \( y = mx + c \) where \( m \) is the slope and \( c \) is the y-intercept. We can rewrite:
\[ f(x) = 2x - \arccos\left(\frac{1}{x}\right) \approx 2x - \frac{\pi}{2} \text{ as } x \to \infty \]

### Step 6: Identify the asymptote
Thus, the asymptote can be identified as:
\[ y = 2x - \frac{\pi}{2} \]

### Conclusion
The asymptote of the curve \( y = 2x - \arccos\left(\frac{1}{x}\right) \) is:
\[ y = 2x - \frac{\pi}{2} \]