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
Here's a detailed analysis of the function f(x) = (e^x)/(x+2). a) The domain of f(x) The function f(x) is defined for all real numbers where the denominator is not zero. The denominator is x+2. Setting the denominator to zero: x+2 = 0 x = -2 Thus, the function is defined for all real numbers except x=-2. The domain is R \-2\. b) The limits at the boundaries of the domain The boundaries of the domain are x -, x -2^-, x -2^+, and x . 1. As x -: _x - f(x) = _x - (e^x)/(x+2) As x -, e^x 0 and x+2 -. _x - f(x) = (0)/(-) = 0 2. As x -2^- (from the left side of -2): _x -2^- f(x) = _x -2^- (e^x)/(x+2) As x -2^-, e^x e^-2 (a positive constant) and x+2 0^- (a small negative number). _x -2^- f(x) = e^-20^- = - 3. As x -2^+ (from the right side of