Let `f(x)` be polynomial of degree 4 with roots 1,2,3,4 and leading coefficient 1 and `g(x)` be the polynomial of degree 4 with roots `1,1/2,1/3` and `1/4` with leading coefficient 1. Then `lim_(xto1)(f(x))/(g(x))` equals

Options

A. (a) `1/24`
B. (b) `-24`
C. (c) `1/12`
D. (d) `-1/12`

Answer

To solve the problem, we need to find the limit:

\[ \lim_{x \to 1} \frac{f(x)}{g(x)} \]

where \( f(x) \) and \( g(x) \) are polynomials defined as follows:

1. \( f(x) \) is a polynomial of degree 4 with roots 1, 2, 3, and 4, and leading coefficient 1.
2. \( g(x) \) is a polynomial of degree 4 with roots \( 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4} \) and leading coefficient 1.

### Step 1: Write the polynomials \( f(x) \) and \( g(x) \)

Since \( f(x) \) has roots 1, 2, 3, and 4, we can express it as:

\[ f(x) = (x - 1)(x - 2)(x - 3)(x - 4) \]

Similarly, for \( g(x) \) with roots \( 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4} \):

\[ g(x) = \left(x - 1\right)\left(x - \frac{1}{2}\right)\left(x - \frac{1}{3}\right)\left(x - \frac{1}{4}\right) \]

### Step 2: Simplify the limit expression

Now we substitute these expressions into the limit:

\[ \lim_{x \to 1} \frac{f(x)}{g(x)} = \lim_{x \to 1} \frac{(x - 1)(x - 2)(x - 3)(x - 4)}{(x - 1)\left(x - \frac{1}{2}\right)\left(x - \frac{1}{3}\right)\left(x - \frac{1}{4}\right)} \]

We can cancel out the \( (x - 1) \) terms in the numerator and denominator:

\[ = \lim_{x \to 1} \frac{(x - 2)(x - 3)(x - 4)}{\left(x - \frac{1}{2}\right)\left(x - \frac{1}{3}\right)\left(x - \frac{1}{4}\right)} \]

### Step 3: Substitute \( x = 1 \)

Now we can substitute \( x = 1 \):

\[ = \frac{(1 - 2)(1 - 3)(1 - 4)}{\left(1 - \frac{1}{2}\right)\left(1 - \frac{1}{3}\right)\left(1 - \frac{1}{4}\right)} \]

Calculating the numerator:

\[ (1 - 2) = -1, \quad (1 - 3) = -2, \quad (1 - 4) = -3 \]

Thus, the numerator becomes:

\[ (-1)(-2)(-3) = -6 \]

Now calculating the denominator:

\[ \left(1 - \frac{1}{2}\right) = \frac{1}{2}, \quad \left(1 - \frac{1}{3}\right) = \frac{2}{3}, \quad \left(1 - \frac{1}{4}\right) = \frac{3}{4} \]

Thus, the denominator becomes:

\[ \frac{1}{2} \cdot \frac{2}{3} \cdot \frac{3}{4} = \frac{1 \cdot 2 \cdot 3}{2 \cdot 3 \cdot 4} = \frac{6}{24} = \frac{1}{4} \]

### Step 4: Final calculation

Now we can compute the limit:

\[ = \frac{-6}{\frac{1}{4}} = -6 \cdot 4 = -24 \]

### Conclusion

Thus, the limit is:

\[ \lim_{x \to 1} \frac{f(x)}{g(x)} = -24 \]

### Answer

The correct answer is \( \boxed{-24} \).