Given that `f(x) = (4-7x)/((3x+4)) , I = Lim_(x to 2 ) f(x) and m = Lim_(x to 0) f(x) ` , form the equation whose are `1/l , 1/m `

Answer

To solve the problem step-by-step, we will follow the instructions given in the video transcript.

### Step 1: Identify the function and limits
Given the function:
\[ f(x) = \frac{4 - 7x}{3x + 4} \]

We need to find:
- \( L = \lim_{x \to 2} f(x) \)
- \( m = \lim_{x \to 0} f(x) \)

### Step 2: Calculate \( L \)
To find \( L \):
\[ L = \lim_{x \to 2} f(x) = \lim_{x \to 2} \frac{4 - 7x}{3x + 4} \]

Substituting \( x = 2 \):
\[ L = \frac{4 - 7(2)}{3(2) + 4} = \frac{4 - 14}{6 + 4} = \frac{-10}{10} = -1 \]

Thus,
\[ L = -1 \]

### Step 3: Calculate \( m \)
To find \( m \):
\[ m = \lim_{x \to 0} f(x) = \lim_{x \to 0} \frac{4 - 7x}{3x + 4} \]

Substituting \( x = 0 \):
\[ m = \frac{4 - 7(0)}{3(0) + 4} = \frac{4}{4} = 1 \]

Thus,
\[ m = 1 \]

### Step 4: Find the roots \( \frac{1}{L} \) and \( \frac{1}{m} \)
Now, we need to find:
- \( \frac{1}{L} = \frac{1}{-1} = -1 \)
- \( \frac{1}{m} = \frac{1}{1} = 1 \)

### Step 5: Form the equation with roots \( -1 \) and \( 1 \)
The equation whose roots are \( -1 \) and \( 1 \) can be formed using the fact that if \( r_1 \) and \( r_2 \) are roots, the equation can be expressed as:
\[ (y - r_1)(y - r_2) = 0 \]

Substituting the roots:
\[ (y + 1)(y - 1) = 0 \]

Expanding this:
\[ y^2 - 1 = 0 \]

### Final Equation
Thus, the required equation is:
\[ y^2 - 1 = 0 \]