Find the constant a, b and c such that `lim_(x->0)(axe^x-blog(1+x)+cxe^(-x))/(x^2sinx) = 2`

Answer

To solve the problem, we need to find the constants \( a \), \( b \), and \( c \) such that

\[ \lim_{x \to 0} \frac{a x e^x - b \log(1+x) + c x e^{-x}}{x^2 \sin x} = 2. \]

### Step 1: Simplify the expression

As \( x \to 0 \), we can use the approximations:
- \( \sin x \approx x \)
- \( \log(1+x) \approx x \)

Thus, we can rewrite the limit as:

\[ \lim_{x \to 0} \frac{a x e^x - b x + c x e^{-x}}{x^2 x} = \lim_{x \to 0} \frac{a x e^x - b x + c x e^{-x}}{x^3}. \]

### Step 2: Factor out \( x \)

Now, we can factor \( x \) out from the numerator:

\[ = \lim_{x \to 0} \frac{x(a e^x - b + c e^{-x})}{x^3} = \lim_{x \to 0} \frac{a e^x - b + c e^{-x}}{x^2}. \]

### Step 3: Apply L'Hôpital's Rule

Since both the numerator and denominator approach 0 as \( x \to 0 \), we can apply L'Hôpital's Rule:

\[ = \lim_{x \to 0} \frac{a e^x - c e^{-x}}{2x}. \]

### Step 4: Apply L'Hôpital's Rule again

Again, both the numerator and denominator approach 0, so we apply L'Hôpital's Rule a second time:

\[ = \lim_{x \to 0} \frac{a e^x + c e^{-x}}{2}. \]

### Step 5: Evaluate the limit

Now, substituting \( x = 0 \):

\[ = \frac{a + c}{2}. \]

### Step 6: Set the limit equal to 2

To satisfy the original condition, we set:

\[ \frac{a + c}{2} = 2. \]

Multiplying through by 2 gives:

\[ a + c = 4. \quad \text{(Equation 1)} \]

### Step 7: Find conditions from the first derivative

From the first application of L'Hôpital's Rule, we also need:

\[ a - c = 0. \quad \text{(Equation 2)} \]

### Step 8: Solve the equations

From Equation 2, we have:

\[ a = c. \]

Substituting \( c \) with \( a \) in Equation 1:

\[ a + a = 4 \implies 2a = 4 \implies a = 2. \]

### Step 9: Find \( c \)

Since \( a = c \), we have:

\[ c = 2. \]

### Step 10: Find \( b \)

Now, we substitute \( a \) and \( c \) back into Equation 1 to find \( b \):

\[ a + c = b \implies 2 + 2 = b \implies b = 4. \]

### Final Result

Thus, the values of \( a \), \( b \), and \( c \) are:

\[ a = 2, \quad b = 4, \quad c = 2. \]