The value of `lim_(x->pi/4)(1+[x])^(1//ln(tanx))` (where[.] denote the greatest integer function) is equal to

Options

A. 0
B. 1
C. e
D. `1/e`

Answer

To solve the limit \( \lim_{x \to \frac{\pi}{4}} \left( 1 + [x] \right)^{\frac{1}{\ln(\tan x)}} \), where \([x]\) denotes the greatest integer function, we can follow these steps:

### Step 1: Evaluate the greatest integer function \([x]\) as \(x\) approaches \(\frac{\pi}{4}\)

As \(x\) approaches \(\frac{\pi}{4}\) (which is approximately \(0.785\)), the greatest integer function \([x]\) will equal \(0\) because \([0.785] = 0\).

### Step 2: Substitute \([x]\) into the limit expression

Now, substituting \([x] = 0\) into the limit expression gives us:

\[ \lim_{x \to \frac{\pi}{4}} \left( 1 + 0 \right)^{\frac{1}{\ln(\tan x)}} = \lim_{x \to \frac{\pi}{4}} 1^{\frac{1}{\ln(\tan x)}} \]

### Step 3: Simplify the expression

Since \(1\) raised to any power is still \(1\), we have:

\[ 1^{\frac{1}{\ln(\tan x)}} = 1 \]

### Step 4: Analyze \(\ln(\tan x)\) as \(x\) approaches \(\frac{\pi}{4}\)

Next, we need to consider what happens to \(\ln(\tan x)\) as \(x\) approaches \(\frac{\pi}{4}\). We know that:

\[ \tan\left(\frac{\pi}{4}\right) = 1 \implies \ln(\tan\left(\frac{\pi}{4}\right)) = \ln(1) = 0 \]

### Step 5: Determine the limit

Thus, we are evaluating:

\[ \lim_{x \to \frac{\pi}{4}} 1^{\frac{1}{\ln(\tan x)}} \]

As \(x\) approaches \(\frac{\pi}{4}\), \(\ln(\tan x)\) approaches \(0\), leading to an indeterminate form of \(1^{\infty}\). To resolve this, we can rewrite the limit using the exponential function.

### Step 6: Rewrite the limit using the exponential function

Using the property that \(a^b = e^{b \ln a}\), we can rewrite the limit as:

\[ \lim_{x \to \frac{\pi}{4}} e^{\frac{1}{\ln(\tan x)} \ln(1 + [x])} \]

Since \([x] = 0\), we have:

\[ \ln(1 + [x]) = \ln(1) = 0 \]

### Step 7: Final evaluation

Thus, we have:

\[ \lim_{x \to \frac{\pi}{4}} e^{\frac{1}{\ln(\tan x)} \cdot 0} = e^0 = 1 \]

### Conclusion

The value of the limit is:

\[ \boxed{1} \]