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 are the solutions to the problems. 1. Step 1: Substitute x=1 into the expression. (1-1)/(1-1) = (0)/(0) This is an indeterminate form. Step 2: Simplify the expression. For x ≠ 1, the expression (x-1)/(x-1) simplifies to 1. Step 3: Evaluate the limit. _x 1 (x-1)/(x-1) = _x 1 1 = 1 The limit is 1. 2. Step 1: Substitute x=1 into the expression. (1^2-1)/(1^2-1) = (1-1)/(1-1) = (0)/(0) This is an indeterminate form. Step 2: Simplify the expression. For x ≠ 1, the expression (x^2-1)/(x^2-1) simplifies to 1. Step 3: Evaluate the limit. _x 1 (x^2-1)/(x^2-1) = _x 1 1 = 1 The limit is 1. 3. Step 1: Substitute x=3 into the expression. (3^3-27)/(3-9) = (27-27)/(-6) = (0)/(-6) = 0 This is not an indeterminate form. Step 2: Evaluate the limit. _x 3 (x^3-27)/(x-9) = 0 The limit is 0. 4. Step 1: Substitute x=0 into the expression. ((3+0)^2-9)/(0) = (3^2-9)/(0) = (9-9)/(0) = (0)/(0) This is an indeterminate form. Step 2: Expand the numerator. (3+x)^2 - 9 = (9 + 6x + x^2) - 9 = 6x + x^2 Step 3: Simplify the expression. (6x+x^2)/(x) = (x(6+x))/(x) = 6+x for x ≠ 0 Step 4: Evaluate the limit. _x 0 (6+x) = 6+0 = 6 The limit is 6. 5. Step 1: Substitute x=1 into the expression. (1^3-1)/(1^2-3(1)+2) = (1-1)/(1-3+2) = (0)/(0) This is an indeterminate form. Step 2: Factor the numerator and denominator. The numerator is a difference of cubes: x^3-1 = (x-1)(x^2+x+1). The denominator is a quadratic: x^2-3x+2 = (x-1)(x-2). Step 3: Simplify the expression. ((x-1)(x^2+x+1))/((x-1)(x-2)) = (x^2+x+1)/(x-2) for x ≠ 1 Step 4: Evaluate the limit. _x 1 (x^2+x+1)/(x-2) = (1^2+1+1)/(1-2) = (3)/(-1) = -3 The limit is -3. 6. Step 1: Substitute x=5 into the expression. (5^2-7(5)+10)/(5^2-25) = (25-35+10)/(25-25) = (0)/(0) This is an indeterminate form. Step 2: Factor the numerator and denominator. The numerator is a quadratic: x^2-7x+10 = (x-2)(x-5). The denominator is a difference of squares: x^2-25 = (x-5)(x+5). Step 3: Simplify the expression. ((x-2)(x-5))/((x-5)(x+5)) = (x-2)/(x+5) for x ≠ 5 Step 4: Evaluate the limit. _x 5 (x-2)/(x+5) = (5-2)/(5+5) = (3)/(10) The limit is (3)/(10). For questions 7-11, we will find the derivative (dy)/(dx). 7. Step 1: Rewrite the function. y = (1)/( x) = x Step 2: Differentiate y with respect to x. (dy)/(dx) = (d)/(dx)( x) = x x The derivative is x x. 8. Step 1: Rewrite the function. y = (1)/( x) = x Step 2: Differentiate y with respect to x. (dy)/(dx) = (d)/(dx)( x) = - x x The derivative is - x x. 9. Step 1: Apply the chain rule. Let u = x^2+4. Then y = u^4. (dy)/(dx) = (dy)/(du) · (du)/(dx) Step 2: Find (dy)/(du) and (du)/(dx). (dy)/(du) = (d)/(du)(u^4) = 4u^3 (du)/(dx) = (d)/(dx)(x^2+4) = 2x Step 3: Substitute back u = x^2+4. (dy)/(dx) = 4(x^2+4)^3 · (2x) = 8x(x^2+4)^3 The derivative is 8x(x^2+4)^3. 10. Step 1: Differentiate y with respect to x. (dy)/(dx) = (d)/(dx)( x) = x The derivative is x. 11. Step 1: Apply the chain rule. Let u = x^2+4. Then y = u^3. (dy)/(dx) = (dy)/(du) · (du)/(dx) Step 2: Find (dy)/(du) and (du)/(dx). (dy)/(du) = (d)/(du)(u^3) = 3u^2 (du)/(dx) = (d)/(dx)(x^2+4) = 2x Step 3: Substitute back u = x^2+4. (dy)/(dx) = 3(x^2+4)^2 · (2x) = 6x(x^2+4)^2 The derivative is 6x(x^2+4)^2. Send me the next one 📸