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
Fresh day ThegirlGodishelping🥰💙🫠, let's solve. To evaluate the limit _x 0^+ (1 + 3x)^(1)/(2x), we first identify its form. As x 0^+, the base (1 + 3x) approaches 1, and the exponent (1)/(2x) approaches . This is an indeterminate form of type 1^. We can use the standard limit identity: _u 0 (1 + u)^(1)/(u) = e Step 1: Substitute to match the standard form. Let u = 3x. As x 0^+, u 0^+. We need the exponent to be in terms of u. The original exponent is (1)/(2x). Since u = 3x, we have x = (u)/(3). Substitute x = (u)/(3) into the exponent: (1)/(2x) = (1)/(2(u)3) = (1)/(2u)3 = (3)/(2u) Step 2: Rewrite the limit expression using the substitution. _x 0^+ (1 + 3x)^(1)/(2x) = _u 0^+ (1 + u)^(3)/(2u) Step 3: Manipulate the exponent to isolate the standard form. We can rewrite the exponent (3)/(2u) as (1)/(u) · (3)/(2). _u 0^+ (1 + u)^(1)/(u) · (3)/(2) Using the exponent rule (a^b)^c = a^bc, we can write this as: _u 0^+ ( (1 + u)^(1)/(u) )^(3)/(2) Step 4: Apply the standard limit and simplify. Since _u 0^+ (1 + u)^(1)/(u) = e, we substitute this into the expression: e^(3)/(2) The final answer is e^(3)/(2). Send me the next one 📸