The value `lim_(x to tan^(-1) 3) (tan^6 x- 2tan^5 x - 3tan^4 x)/(tan^2 x -4 tan x+3)`

Answer

To evaluate the limit

\[ \lim_{x \to \tan^{-1}(3)} \frac{\tan^6 x - 2\tan^5 x - 3\tan^4 x}{\tan^2 x - 4\tan x + 3}, \]

we will follow these steps:

### Step 1: Substitute \( t = \tan x \)

Since \( x \to \tan^{-1}(3) \), we have \( t \to 3 \). Thus, we can rewrite the limit in terms of \( t \):

\[ \lim_{t \to 3} \frac{t^6 - 2t^5 - 3t^4}{t^2 - 4t + 3}. \]

### Step 2: Factor the numerator and denominator

First, we will factor the numerator \( t^6 - 2t^5 - 3t^4 \):

1. Factor out \( t^4 \):

\[ t^4(t^2 - 2t - 3). \]

2. Now, we need to factor \( t^2 - 2t - 3 \). This can be factored as:

\[ (t - 3)(t + 1). \]

So, the numerator becomes:

\[ t^4(t - 3)(t + 1). \]

Next, we factor the denominator \( t^2 - 4t + 3 \):

1. This can be factored as:

\[ (t - 3)(t - 1). \]

### Step 3: Rewrite the limit

Now we can rewrite the limit as:

\[ \lim_{t \to 3} \frac{t^4(t - 3)(t + 1)}{(t - 3)(t - 1)}. \]

### Step 4: Cancel common factors

We can cancel the common factor \( (t - 3) \) from the numerator and the denominator:

\[ \lim_{t \to 3} \frac{t^4(t + 1)}{t - 1}. \]

### Step 5: Substitute \( t = 3 \)

Now, we substitute \( t = 3 \):

\[ \frac{3^4(3 + 1)}{3 - 1} = \frac{81 \cdot 4}{2} = \frac{324}{2} = 162. \]

### Final Answer

Thus, the value of the limit is

\[ \boxed{162}. \]