Final answer:
The limit of sin(9x)/sin(3x) as x approaches 0 is evaluated using the standard trigonometric limit sin(x)/x as x approaches 0, resulting in the value 3.
Step-by-step explanation:
The student is asking to evaluate the limit of the function as x approaches zero. The function in question is the sine of 9x divided by the sine of 3x. This is a common type of limit problem that can be solved using L'Hôpital's Rule or the properties of trigonometric functions, specifically exploiting the fact that sin(θ)/θ approaches 1 as θ approaches zero.
To solve this limit, we can use the fact that for small angles (such as when x approaches zero), the sine of an angle is approximately equal to the angle itself (when measured in radians). Therefore, the limit can be rewritten as:
lim _{x\rightarrow 0} \frac{sin(9x)}{sin(3x)} = lim _{x\rightarrow 0} \frac{9x}{3x} = lim _{x\rightarrow 0} 3 = 3.
The limit of sin(9x)/sin(3x) is simply 3, borrowing the property of sine for small angles. It's also worth noting that if we had to use L'Hôpital's Rule, we would differentiate the numerator and the denominator until we can evaluate the limit directly at x = 0.