Question

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Find the limit, if it exists.
lim _x →[infinity]sin (3 x)/x

Answer 1

The limit of sin(3x)/x as x approaches infinity is 0.

Step-by-step explanation:

The given limit is limx →∞ sin(3x)/x. To find the limit, we can use L'Hôpital's Rule. Differentiating both the numerator and the denominator, we get (3cos(3x))/1. Since the limit as x approaches infinity is a certain type of indeterminate form (0/0), we need to apply L'Hôpital's Rule again. Differentiating once more, we get -9sin(3x)/1. Now, as x approaches infinity, sin(3x) oscillates between -1 and 1, so -9sin(3x) approaches a value between -9 and 9. Therefore, the limit of sin(3x)/x as x approaches infinity is 0.