![Evaluate the limit: lim (x->0) [tan(3x) - 3tan(x)] / x3](/_next/image?url=https%3A%2F%2Fohijasnjvprzodcmxltt.supabase.co%2Fstorage%2Fv1%2Fobject%2Fpublic%2Fwhatsapp-images%2F1778146850014-9d5269afc930e732.png&w=3840&q=75)
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Okay, ~Mataa🫶🏽, let's tackle the second limit. First, substitute x=0 into the expression to check for an indeterminate form: ((3 · 0) - 3 (0))/(0^3) = ((0) - 3 (0))/(0) = (0 - 0)/(0) = (0)/(0) This is an indeterminate form, so we need to manipulate the expression. Step 1: Use the triple angle identity for tangent, (3x) = (3 x - ^3 x)/(1 - 3 ^2 x). Substitute this into the limit expression: _x 0 (3 x - ^3 x)/(1 - 3 ^2 x) - 3 xx^3 Step 2: Combine the terms in the numerator by finding a common denominator. _x 0 (3 x - ^3 x - 3 x (1 - 3 ^2 x))/(x^3 (1 - 3 ^2 x)) _x 0 (3 x - ^3 x - 3 x + 9 ^3 x)/(x^3 (1 - 3 ^2 x)) Step 3: Simplify the numerator. _x 0 (8 ^3 x)/(x^3 (1 - 3 ^2 x)) Step 4: Rearrange the terms to utilize the standard limit _x 0 ( x)/(x) = 1. _x 0 ( 8 · ( ( x)/(x) )^3 · (1)/(1 - 3 ^2 x) ) Step 5: Evaluate each part of the limit. ( _x 0 8 ) · ( _x 0 ( ( x)/(x) )^3 ) · ( _x 0 (1)/(1 - 3 ^2 x) ) = 8 · (1)^3 · (1)/(1 - 3 ^2(0)) = 8 · 1 · (1)/(1 - 3 · 0^2) = 8 · 1 · (1)/(1 - 0) = 8 · 1 · 1 = 8 The limit exists and its value is: 8 That's 2 down. 3 left today — send the next one.