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 evaluate these limits without L'Hôpital's Rule. For the first limit: _x 0 6e^3x - 6e^x - 1 First, substitute x=0: 6e^3 · 0 - 6e^0 - 1 = (6 · 1 - 6)/(1 - 1) = (0)/(0) This is an indeterminate form. We will use the standard limit _x 0 e^ax - 1x = a. Step 1: Factor out 6 from the numerator. _x 0 6(e^3x - 1)e^x - 1 Step 2: Multiply the numerator and denominator by x to create the standard limit forms. _x 0 6 · e^3x - 1x · (x)/(e^x - 1) Step 3: Rearrange the terms to clearly show the standard limits. _x 0 6 · e^3x - 1x · (1)/(e^x - 1)x Step 4: Apply the standard limit _x 0 e^ax - 1x = a. 6 · ( _x 0 e^3x - 1x ) · ( _x 0 (1)/(e^x - 1)x ) = 6 · (3) · (1)/(1) = 18 The first limit is: 18 For the second limit: _x 0 e^4x - e^2xx First, substitute x=0: e^4 · 0 - e^2 · 00 = (e^0 - e^0)/(0) = (1 - 1)/(0) = (0)/(0) This is an indeterminate form. We will again use the standard limit _x 0 e^ax - 1x = a. Step 1: Add and subtract 1 in the numerator to create terms that fit the standard limit form. _x 0 e^4x - 1 - e^2x + 1x Step 2: Group the terms and split the fraction. _x 0 ( e^4x - 1x - e^2x - 1x ) Step 3: Apply the limit linearity property. _x 0 e^4x - 1x - _x 0 e^2x - 1x Step 4: Apply the standard limit _x 0 e^ax - 1x = a to each term. = 4 - 2 = 2 The second limit is: 2 3 done, 2 left today. You're making progress.