Let `lim_(xto1)(x^(a)-ax+a-1)/((x-1)^(2))=f(a)`. The value of `f(101)` equals

Options

A. 5050
B. 5151
C. 4950
D. 101

Answer

To solve the limit problem given by

\[ \lim_{x \to 1} \frac{x^a - ax + a - 1}{(x - 1)^2} = f(a), \]

we will follow these steps:

### Step 1: Substitute \( x = 1 + h \)

Let \( x = 1 + h \) where \( h \to 0 \) as \( x \to 1 \). Then, we rewrite the limit:

\[ \lim_{h \to 0} \frac{(1 + h)^a - a(1 + h) + a - 1}{(h)^2}. \]

### Step 2: Expand \( (1 + h)^a \) using the binomial series

Using the binomial expansion for \( (1 + h)^a \):

\[ (1 + h)^a = 1 + ah + \frac{a(a-1)}{2}h^2 + O(h^3). \]

### Step 3: Substitute the expansion back into the limit

Now substitute this expansion into the limit expression:

\[ \lim_{h \to 0} \frac{\left(1 + ah + \frac{a(a-1)}{2}h^2 + O(h^3)\right) - a(1 + h) + a - 1}{h^2}. \]

### Step 4: Simplify the expression

Now simplify the numerator:

\[ 1 + ah + \frac{a(a-1)}{2}h^2 + O(h^3) - (a + ah) + a - 1. \]

This simplifies to:

\[ ah - ah + \frac{a(a-1)}{2}h^2 + O(h^3) = \frac{a(a-1)}{2}h^2 + O(h^3). \]

### Step 5: Divide by \( h^2 \)

Now, divide the entire expression by \( h^2 \):

\[ \lim_{h \to 0} \left(\frac{a(a-1)}{2} + \frac{O(h^3)}{h^2}\right). \]

As \( h \to 0 \), the term \( \frac{O(h^3)}{h^2} \to 0 \). Thus, we have:

\[ f(a) = \frac{a(a-1)}{2}. \]

### Step 6: Find \( f(101) \)

Now, we need to find \( f(101) \):

\[ f(101) = \frac{101(101 - 1)}{2} = \frac{101 \times 100}{2} = \frac{10100}{2} = 5050. \]

### Final Answer

Thus, the value of \( f(101) \) is

\[ \boxed{5050}. \]