Evaluate the limit: lim (x -> 0) (tan(3x)/x)

₹0.0

Question: Evaluate the limit: lim (x -> 0) (tan(3x)/x)

Options:

Correct Answer: 3

Solution:

Using the standard limit lim (x -> 0) (tan(x)/x) = 1, we have lim (x -> 0) (tan(3x)/x) = 3 * lim (x -> 0) (tan(3x)/(3x)) = 3 * 1 = 3.

Evaluate the limit: lim (x -> 0) (tan(3x)/x)

Evaluate the limit: lim (x -> 0) (tan(3x)/x)

Step 1: Identify the limit we want to evaluate: lim (x -> 0) (tan(3x)/x).
Step 2: Recognize a useful standard limit: lim (x -> 0) (tan(x)/x) = 1.
Step 3: Rewrite the limit using the standard limit. We can express tan(3x) in terms of tan(3x)/(3x): lim (x -> 0) (tan(3x)/x) = lim (x -> 0) (tan(3x)/(3x)) * 3.
Step 4: Now, we can evaluate lim (x -> 0) (tan(3x)/(3x)). Since 3x approaches 0 as x approaches 0, we can use the standard limit: lim (x -> 0) (tan(3x)/(3x)) = 1.
Step 5: Substitute this result back into our expression: lim (x -> 0) (tan(3x)/x) = 3 * 1.
Step 6: Finally, calculate the result: 3 * 1 = 3.

Limit Evaluation – Understanding how to evaluate limits, particularly using standard limit results.
Trigonometric Limits – Applying the standard limit of tan(x)/x as x approaches 0.
Constant Multiplication in Limits – Recognizing how to handle constants when evaluating limits involving functions.