Find the value of `lim_(x to oo) (pi/4 - "tan"^(-1) (x+1)/(x+2))` .

Answer

To find the value of the limit \( \lim_{x \to \infty} \left( \frac{\pi}{4} - \tan^{-1} \left( \frac{x+1}{x+2} \right) \right) \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the limit:
\[ \lim_{x \to \infty} \left( \frac{\pi}{4} - \tan^{-1} \left( \frac{x+1}{x+2} \right) \right) \]

### Step 2: Simplify the argument of the arctangent
We simplify \( \frac{x+1}{x+2} \):
\[ \frac{x+1}{x+2} = \frac{x(1 + \frac{1}{x})}{x(1 + \frac{2}{x})} = \frac{1 + \frac{1}{x}}{1 + \frac{2}{x}} \]

### Step 3: Take the limit of the argument as \( x \to \infty \)
As \( x \to \infty \):
\[ \frac{1 + \frac{1}{x}}{1 + \frac{2}{x}} \to \frac{1 + 0}{1 + 0} = 1 \]

### Step 4: Substitute back into the arctangent
Now we substitute this limit back into the arctangent:
\[ \tan^{-1} \left( \frac{x+1}{x+2} \right) \to \tan^{-1}(1) \]

### Step 5: Evaluate the arctangent
We know that:
\[ \tan^{-1}(1) = \frac{\pi}{4} \]

### Step 6: Substitute back into the limit expression
Now we substitute this value back into our limit:
\[ \lim_{x \to \infty} \left( \frac{\pi}{4} - \tan^{-1}(1) \right) = \frac{\pi}{4} - \frac{\pi}{4} = 0 \]

### Final Answer
Thus, the value of the limit is:
\[ \boxed{0} \]