Question

Find the limit lim(x → 0) (sin(3x)sin(5x))/(x²).

A. 15
B. 5
C. 1/2
D. 0

Answer 1

The limit of (sin(3x)sin(5x))/(x²) as x approaches 0 is equal to 15. The correct answer is option A .

Step-by-step explanation:

To find the limit lim(x → 0) (sin(3x)sin(5x))/(x²), we can use L'Hospital's Rule.

1. First, differentiate the numerator and denominator with respect to x. The derivative of sin(3x) is 3cos(3x), and the derivative of sin(5x) is 5cos(5x). The derivative of x² is 2x.
2. Now, substitute x = 0 into the derivatives. We get 3cos(0) * 5cos(0) / (2*0) = 15 / 0 = undefiend.
3. Since the limit is undefined, we cannot evaluate it using L'Hospital's Rule. However, notice that sin(3x) and sin(5x) are both 0 when x = 0, and x² is also 0 when x = 0. Therefore, we can simplify the expression as sin(3x)sin(5x)/x² = 0/0.
4. To find the limit in this case, we can use the properties of sin function. For small angles, sin is approximately equal to the angle in radians. So, as x approaches 0, sin(3x) is approximately equal to 3x, and sin(5x) is approximately equal to 5x. Thus, the limit becomes lim(x → 0) (3x * 5x) / (x²) = lim(x → 0) 15x² / x² = lim(x → 0) 15.

Therefore, the limit lim(x → 0) (sin(3x)sin(5x))/(x²) is equal to 15.