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
To solve the limit _x 1 sqrt(x+2)-1x-1: Step 1: Evaluate the expression by direct substitution. Substitute x=1 into the numerator and the denominator: Numerator: sqrt(1+2)-1 = sqrt(3)-1 Denominator: 1-1 = 0 The expression takes the form sqrt(3)-10. Since sqrt(3) ≈ 1.732, the numerator sqrt(3)-1 ≈ 0.732, which is a positive non-zero value. Step 2: Analyze the one-sided limits. Since the numerator approaches a non-zero value and the denominator approaches zero, the limit will be either +, -, or it will not exist. We need to examine the sign of the denominator as x approaches 1 from both sides. • As x 1^+ (x approaches 1 from values greater than 1): The denominator x-1 will be positive. The numerator sqrt(x+2)-1 is positive (as sqrt(3)-1 > 0). So, _x 1^+ sqrt(x+2)-1x-1 = positivepositive = +. • As x 1^- (x approaches 1 from values less than 1): The denominator x-1 will be negative. The numerator sqrt(x+2)-1 is positive. So, _x 1^- sqrt(x+2)-1x-1 = positivenegative = -. Step 3: Conclude the limit. Since the left-hand limit (-) and the right-hand limit (+) are not equal, the overall limit does not exist. _x 1 sqrt(x+2)-1x-1 = Does not exist Send me the next one 📸