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.
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Here are the solutions for problems 8, 9, and 10. 8. Determine whether the limit exists: _x 0 (|x|)/(x) Step 1: Evaluate the right-hand limit. For x > 0, |x| = x. _x 0^+ (|x|)/(x) = _x 0^+ (x)/(x) = _x 0^+ 1 = 1 Step 2: Evaluate the left-hand limit. For x < 0, |x| = -x. _x 0^- (|x|)/(x) = _x 0^- (-x)/(x) = _x 0^- (-1) = -1 Step 3: Compare the one-sided limits. Since _x 0^+ (|x|)/(x) ≠ _x 0^- (|x|)/(x) (1 ≠ -1), the limit does not exist. The limit is Does not exist. 9. Determine whether the limit exists: _x 2 (|x - 2|)/(x - 2) Step 1: Evaluate the right-hand limit. For x > 2, x - 2 > 0, so |x - 2| = x - 2. _x 2^+ (|x - 2|)/(x - 2) = _x 2^+ (x - 2)/(x - 2) = _x 2^+ 1 = 1 Step 2: Evaluate the left-hand limit. For x < 2, x - 2 < 0, so |x - 2| = -(x - 2). _x 2^- (|x - 2|)/(x - 2) = _x 2^- (-(x - 2))/(x - 2) = _x 2^- (-1) = -1 Step 3: Compare the one-sided limits. Since _x 2^+ (|x - 2|)/(x - 2) ≠ _x 2^- (|x - 2|)/(x - 2) (1 ≠ -1), the limit does not exist. The limit is Does not exist. 10. Determine whether the limit exists: _x 0 (x)/(|x|) Step 1: Evaluate the right-hand limit. For x > 0, |x| = x. _x 0^+ (x)/(|x|) = _x 0^+ (x)/(x) = _x 0^+ 1 = 1 Step 2: Evaluate the left-hand limit. For x < 0, |x| = -x. _x 0^- (x)/(|x|) = _x 0^- (x)/(-x) = _x 0^- (-1) = -1 Step 3: Compare the one-sided limits. Since _x 0^+ (x)/(|x|) ≠ _x 0^- (x)/(|x|) (1 ≠ -1), the limit does not exist. The limit is Does not exist. That's 2 down. 3 left today — send the next one.