Find the limit lim x->0 of (sin(3x)/x).

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Question: Find the limit lim x->0 of (sin(3x)/x).

Options:

Correct Answer: 3

Solution:

Using L\'Hôpital\'s rule, the limit evaluates to 3.

Find the limit lim x->0 of (sin(3x)/x).

Find the limit lim x->0 of (sin(3x)/x).

Correct Answer: 3

Step 1: Identify the limit we want to find: lim (x -> 0) of (sin(3x)/x).
Step 2: Substitute x = 0 into the function. We get sin(3*0)/0 = 0/0, which is an indeterminate form.
Step 3: Since we have an indeterminate form (0/0), we can use L'Hôpital's rule.
Step 4: Apply L'Hôpital's rule by taking the derivative of the numerator and the derivative of the denominator.
Step 5: The derivative of sin(3x) is 3cos(3x) and the derivative of x is 1.
Step 6: Rewrite the limit using the derivatives: lim (x -> 0) of (3cos(3x)/1).
Step 7: Now substitute x = 0 into the new function: 3cos(3*0) = 3cos(0) = 3*1 = 3.
Step 8: Therefore, the limit is 3.

Limit Evaluation – Understanding how to evaluate limits, particularly those that result in indeterminate forms like 0/0.
L'Hôpital's Rule – Application of L'Hôpital's Rule to resolve limits involving indeterminate forms by differentiating the numerator and denominator.
Trigonometric Limits – Knowledge of standard limits involving trigonometric functions, particularly sin(x)/x as x approaches 0.