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
Right athanasie, let's go. b) Evaluate _t 0 ( 3t)/(t) Step 1: Recognize the standard limit form. The limit is of the form _x 0 ( ax)/(x) = a. Step 2: Apply the standard limit. In this case, a=3. _t 0 ( 3t)/(t) = 3 The final answer is 3. c) Evaluate _x ^-1((x-1)/(2x)) Step 1: Evaluate the limit of the argument inside the inverse cosine function. Let L = _x (x-1)/(2x). To evaluate this limit, divide both the numerator and the denominator by the highest power of x, which is x: L = _x (x)/(x) - (1)/(x)(2x)/(x) L = _x (1 - 1)/(x)2 As x , (1)/(x) 0. L = (1 - 0)/(2) = (1)/(2) Step 2: Substitute the limit of the argument into the inverse cosine function. Since ^-1(y) is a continuous function for y [-1, 1], we can write: _x ^-1((x-1)/(2x)) = ^-1(_x (x-1)/(2x)) = ^-1((1)/(2)) Step 3: Find the value of ^-1((1)/(2)). The value y such that y = (1)/(2) in the principal range [0, ] is ()/(3). ^-1((1)/(2)) = ()/(3) The final answer is ()/(3). Got more? Send 'em.