`lim_(xtooo) {(x+5)tan^(-1)(x+5)-(x+1)tan^(-1)(x+1)}` is equal to

Explanation

To solve the limit problem \( \lim_{x \to \infty} \left( (x+5) \tan^{-1}(x+5) - (x+1) \tan^{-1}(x+1) \right) \), we can follow these steps:

### Step 1: Identify the limit
We need to evaluate the limit as \( x \) approaches infinity for the expression given.

### Step 2: Substitute the limit into the expression
As \( x \to \infty \), both \( \tan^{-1}(x+5) \) and \( \tan^{-1}(x+1) \) approach \( \frac{\pi}{2} \). Therefore, we can rewrite the limit:
\[ \lim_{x \to \infty} \left( (x+5) \tan^{-1}(x+5) - (x+1) \tan^{-1}(x+1) \right) \]

### Step 3: Replace \( \tan^{-1}(x+5) \) and \( \tan^{-1}(x+1) \) with \( \frac{\pi}{2} \)
Substituting \( \tan^{-1}(x+5) \) and \( \tan^{-1}(x+1) \) with \( \frac{\pi}{2} \):
\[ = \lim_{x \to \infty} \left( (x+5) \cdot \frac{\pi}{2} - (x+1) \cdot \frac{\pi}{2} \right) \]

### Step 4: Simplify the expression
Now, we can factor out \( \frac{\pi}{2} \):
\[ = \lim_{x \to \infty} \left( \frac{\pi}{2} \left( (x+5) - (x+1) \right) \right) \]
This simplifies to:
\[ = \lim_{x \to \infty} \left( \frac{\pi}{2} \left( x + 5 - x - 1 \right) \right) \]
\[ = \lim_{x \to \infty} \left( \frac{\pi}{2} \cdot (5 - 1) \right) \]

### Step 5: Final calculation
Now, we can simplify further:
\[ = \frac{\pi}{2} \cdot 4 = 2\pi \]

### Conclusion
Thus, the limit evaluates to:
\[ \lim_{x \to \infty} \left( (x+5) \tan^{-1}(x+5) - (x+1) \tan^{-1}(x+1) \right) = 2\pi \]

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