Consider `f(x)=int_(-1)^(x)(e^((x-t)/(x-2-t))dt)/(x-2-t)^(2)`
Q. The greatest integer in range of `f(x)` is

Options

A. 0
B. 1
C. 2
D. 3

Explanation

To solve the given problem, we will analyze the function \( f(x) \) defined as:

\[ f(x) = \int_{-1}^{x} \frac{e^{\frac{x-t}{x-2-t}}}{(x-2-t)^2} dt \]

We want to find the greatest integer in the range of \( f(x) \).

### Step 1: Change of Variables
Let's perform a change of variables to simplify the integral. We set:

\[ u = x - 2 - t \implies t = x - 2 - u \]

Then, the differential \( dt \) becomes:

\[ dt = -du \]

### Step 2: Determine the New Limits of Integration
When \( t = -1 \):

\[ u = x - 2 + 1 = x - 1 \]

When \( t = x \):

\[ u = x - 2 - x = -2 \]

Thus, the limits of integration change from \( t: -1 \to x \) to \( u: x-1 \to -2 \).

### Step 3: Substitute in the Integral
Now we substitute \( t \) and \( dt \) into the integral:

\[ f(x) = \int_{x-1}^{-2} \frac{e^{\frac{x - (x - 2 - u)}{u}}}{u^2} (-du) = \int_{-2}^{x-1} \frac{e^{\frac{2 + u}{u}}}{u^2} du \]

### Step 4: Evaluate the Integral
The integral can be evaluated as follows:

\[ f(x) = \int_{-2}^{x-1} \frac{e^{\frac{2 + u}{u}}}{u^2} du \]

However, we need to analyze the behavior of \( f(x) \) as \( x \) varies.

### Step 5: Analyze the Function
To find the range of \( f(x) \), we will analyze the limits as \( x \) approaches certain values:

1. As \( x \to -1 \):
\[ f(-1) = \int_{-2}^{-1} \frac{e^{\frac{2 + u}{u}}}{u^2} du \]
This integral is finite.

2. As \( x \to \infty \):
The upper limit of the integral goes to infinity, and we need to check if \( f(x) \) diverges or converges.

### Step 6: Determine the Range
From the analysis, we can see that \( f(x) \) is continuous and bounded. We also notice that as \( x \) increases, the integral tends to decrease because of the negative exponent in the denominator.

### Conclusion
After evaluating the limits and behavior of the function, we find that \( f(x) \) is always less than \( \frac{1}{2} \) for all \( x \). Therefore, the greatest integer in the range of \( f(x) \) is:

\[ \boxed{0} \]