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.
Questions & Step-by-step Solutions
1 item
Q
Q: Evaluate the limit: lim (x -> 0) (tan(3x)/x)
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.
Steps: 6
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.
