Let `f (x) =3x ^(10) -7x ^(8) +5x^(6) -21 x ^(3) +3x ^(2) -7`
`265 (lim _(htoo) (h ^(4) +3h^(2))/((f(1-h) -f (1))sin5h))=`

Options

A. 1
B. 2
C. 3
D. `-3`

Answer

To solve the limit problem, we will follow a systematic approach.

**Given:**
\[ f(x) = 3x^{10} - 7x^{8} + 5x^{6} - 21x^{3} + 3x^{2} - 7 \]
We need to evaluate:
\[ 265 \lim_{h \to 0} \frac{h^{4} + 3h^{2}}{(f(1-h) - f(1)) \sin(5h)} \]

### Step 1: Simplify the limit expression
First, we can factor \( h^2 \) out of the numerator:
\[ \lim_{h \to 0} \frac{h^2(h^2 + 3)}{(f(1-h) - f(1)) \sin(5h)} \]

### Step 2: Rewrite the limit
We can rewrite the limit as:
\[ 265 \lim_{h \to 0} \frac{h^2 + 3}{(f(1-h) - f(1)) \frac{\sin(5h)}{h^2}} \]

### Step 3: Apply L'Hôpital's Rule
Since both the numerator and denominator approach 0 as \( h \to 0 \), we can apply L'Hôpital's Rule. However, we will first analyze \( f(1-h) - f(1) \).

### Step 4: Find \( f(1) \)
Calculate \( f(1) \):
\[ f(1) = 3(1)^{10} - 7(1)^{8} + 5(1)^{6} - 21(1)^{3} + 3(1)^{2} - 7 \]
\[ = 3 - 7 + 5 - 21 + 3 - 7 = -34 \]

### Step 5: Find \( f(1-h) \)
Next, we need to find \( f(1-h) \):
\[ f(1-h) = 3(1-h)^{10} - 7(1-h)^{8} + 5(1-h)^{6} - 21(1-h)^{3} + 3(1-h)^{2} - 7 \]

### Step 6: Use Taylor Expansion
Using Taylor expansion around \( h = 0 \):
\[ f(1-h) \approx f(1) - f'(1)h + O(h^2) \]
where \( f'(x) \) is the derivative of \( f(x) \).

### Step 7: Find \( f'(x) \)
Differentiate \( f(x) \):
\[ f'(x) = 30x^{9} - 56x^{7} + 30x^{5} - 63x^{2} + 6 \]

### Step 8: Calculate \( f'(1) \)
Now calculate \( f'(1) \):
\[ f'(1) = 30(1)^{9} - 56(1)^{7} + 30(1)^{5} - 63(1)^{2} + 6 \]
\[ = 30 - 56 + 30 - 63 + 6 = -53 \]

### Step 9: Substitute back into the limit
Now we can substitute back into our limit:
\[ f(1-h) - f(1) \approx -(-53)h = 53h \]
Thus, the limit becomes:
\[ 265 \lim_{h \to 0} \frac{h^2 + 3}{53h \sin(5h)} \]

### Step 10: Simplify the limit
Now, we can simplify:
\[ = 265 \lim_{h \to 0} \frac{h^2 + 3}{53h} \cdot \frac{1}{\sin(5h)} \]

Using the fact that \( \lim_{h \to 0} \frac{\sin(5h)}{5h} = 1 \):
\[ = 265 \cdot \frac{1}{53} \cdot \frac{1}{5} \cdot 3 \]

### Step 11: Final Calculation
Calculating the final value:
\[ = \frac{265 \cdot 3}{53 \cdot 5} = \frac{795}{265} = 3 \]

### Final Answer
Thus, the value of the limit is:
\[ \boxed{3} \]