What is the limit: lim (x -> 0) (tan(3x)/x)?

What is the limit: lim (x -> 0) (tan(3x)/x)?

Step 1: Identify the limit we want to find: lim (x -> 0) (tan(3x)/x).
Step 2: Recall a known limit: lim (x -> 0) (tan(x)/x) = 1.
Step 3: We can rewrite our limit using the known limit. Notice that tan(3x) can be expressed as tan(3x)/(3x) multiplied by 3x/x.
Step 4: Rewrite the limit: lim (x -> 0) (tan(3x)/x) = lim (x -> 0) (tan(3x)/(3x)) * 3.
Step 5: Now, we need to find lim (x -> 0) (tan(3x)/(3x)).
Step 6: Since 3x approaches 0 as x approaches 0, we can use the known limit: lim (u -> 0) (tan(u)/u) = 1, where u = 3x.
Step 7: Therefore, lim (x -> 0) (tan(3x)/(3x)) = 1.
Step 8: Substitute this back into our rewritten limit: lim (x -> 0) (tan(3x)/x) = 1 * 3.
Step 9: Finally, we find that lim (x -> 0) (tan(3x)/x) = 3.

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