`"lt"_(n to infty)[tan {(pi-4)/4 +(1+1/n)^a}]^(n)` =

Options

A. `e^(a)`
B. `e^(2a)`
C. `e^(0)`
D. does not exist

Answer

To solve the limit

\[ \lim_{n \to \infty} \left( \tan \left( \frac{\pi - 4}{4} + \left(1 + \frac{1}{n}\right)^a \right) \right)^n, \]

we will follow these steps:

### Step 1: Simplifying the expression inside the tangent
First, we analyze the term inside the tangent function as \( n \) approaches infinity:

\[ \left(1 + \frac{1}{n}\right)^a \to 1 \quad \text{as } n \to \infty. \]

Thus, we can rewrite the limit as:

\[ \lim_{n \to \infty} \tan\left(\frac{\pi - 4}{4} + 1\right)^n. \]

### Step 2: Calculate the value of \(\frac{\pi - 4}{4} + 1\)
Now, we calculate:

\[ \frac{\pi - 4}{4} + 1 = \frac{\pi - 4 + 4}{4} = \frac{\pi}{4}. \]

### Step 3: Substitute back into the limit
Now, we substitute this back into our limit:

\[ \lim_{n \to \infty} \left(\tan\left(\frac{\pi}{4}\right)\right)^n. \]

### Step 4: Evaluate \(\tan\left(\frac{\pi}{4}\right)\)
We know that:

\[ \tan\left(\frac{\pi}{4}\right) = 1. \]

Thus, we have:

\[ \lim_{n \to \infty} 1^n. \]

### Step 5: Final evaluation of the limit
Since \( 1^n = 1 \) for any value of \( n \), we conclude that:

\[ \lim_{n \to \infty} 1^n = 1. \]

### Final Result
Therefore, the final result of the limit is:

\[ \boxed{1}. \]

---