Question

$\lim_{x\rightarrow 2}\frac{\sin x \cos 2 - \cos x \sin 2}{x-2} =$

• A. -1
• B. 1
• C. 4
• D. 2
• E. 0

Hint:

Recognizing angle addition/subtraction formulas can turn a complex-looking limit into a standard one.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
We are asked to evaluate a limit that is in the indeterminate form \( \frac{0}{0} \). The expression in the numerator resembles the sine angle subtraction formula. Alternatively, the limit's structure matches the definition of a derivative.
Step 2: Key Formula or Approach:
Method 1: Using Trigonometric Identity
1. Recognize the angle subtraction formula for sine: \( \sin(A-B) = \sin A \cos B - \cos A \sin B \).
2. Use the standard limit \( \lim_{u \to 0} \frac{\sin u}{u} = 1 \).
Method 2: Using the Definition of the Derivative
1. The derivative of a function \( f(x) \) at a point \( a \) is defined as \( f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x-a} \).
Step 3: Detailed Explanation:
Using Method 1:
The numerator is \( \sin x \cos 2 - \cos x \sin 2 \). This matches the formula for \( \sin(x-2) \).
So the limit becomes:
\[ \lim_{x \to 2} \frac{\sin(x-2)}{x-2} \] This is a standard limit form. Let \( u = x-2 \). As \( x \to 2 \), we have \( u \to 0 \).
The limit transforms into:
\[ \lim_{u \to 0} \frac{\sin u}{u} \] The value of this standard limit is 1.
Using Method 2:
Let's compare the given limit with the definition of the derivative, \( \lim_{x \to a} \frac{f(x) - f(a)}{x-a} \).
By comparison:
- \( a = 2 \)
- \( f(x) = \sin x \)
- \( f(a) = f(2) = \sin 2 \).
The term \( \cos x \sin 2 \) doesn't immediately fit. Wait, let's re-examine the numerator \( \sin x \cos 2 - \cos x \sin 2 \). This is exactly \( \sin(x-2) \). Let \( f(x) = \sin(x-2) \). Then \( f(2) = \sin(2-2) = \sin(0) = 0 \). The limit is \( \lim_{x \to 2} \frac{\sin(x-2) - 0}{x-2} \). This is the definition of the derivative of the function \( f(x) = \sin(x-2) \) evaluated at \( x=2 \). Let's find \( f'(x) \).
\( f'(x) = \frac{d}{dx}(\sin(x-2)) = \cos(x-2) \).
Now evaluate \( f'(2) \):
\( f'(2) = \cos(2-2) = \cos(0) = 1 \).
Step 4: Final Answer:
The value of the limit is 1.