Find the value of k for which the limit of the piecewise function exists and evaluate the given limit using L'Hôpital's Rule.

You're on a roll — here are the solutions to the limit problems. 6. We need to find k such that _x 2 f(x) exists for the piecewise function: f(x) = kx^2 + 1 & x 2 \\ 3x - k & x > 2 For the limit to exist at x=2, the left-hand limit must equal the right-hand limit. Step 1: Calculate the left-hand limit. _x 2^- f(x) = _x 2^- (kx^2 + 1) Substitute x=2: k(2)^2 + 1 = 4k + 1 Step 2: Calculate the right-hand limit. _x 2^+ f(x) = _x 2^+ (3x - k) Substitute x=2: 3(2) - k = 6 - k Step 3: Set the left-hand limit equal to the right-hand limit and solve for k. 4k + 1 = 6 - k 4k + k = 6 - 1 5k = 5 k = 1 The value of k is 1. 7. We need to evaluate the limit: _x 0 (e^x - x - 1)/(x^2) Step 1: Substitute x=0 into the expression. (e^0 - 0 - 1)/(0^2) = (1 - 0 - 1)/(0) = (0)/(0) This is an indeterminate form, so we apply L'Hôpital's Rule. Step 2: Apply L'Hôpital's Rule by differentiating the numerator and denominator. _x 0 (d)/(dx)(e^x - x - 1)(d)/(dx)(x^2) = _x 0 (e^x - 1)/(2x) Step 3: Substitute x=0 again. (e^0 - 1)/(2(0)) = (1 - 1)/(0) = (0)/(0) This is still an indeterminate form, so we apply L'Hôpital's Rule again. Step 4: Apply L'Hôpital's Rule a second time. _x 0 (d)/(dx)(e^x - 1)(d)/(dx)(2x) = _x 0 (e^x)/(2) Step 5: Substitute x=0 into the simplified expression. (e^0)/(2) = (1)/(2) The limit is (1)/(2). 8. We need to evaluate the limit: _x (1 + (3)/(x))^2x Step 1: Recognize this as a standard limit form. The limit is of the form _x (1 + (a)/(x))^bx = e^ab. Step 2: Identify a and b from the given limit. In this case, a=3 and b=2. Step 3: Apply the formula. e^(3)(2) = e^6 The limit is e^6. 9. We need to evaluate the limit: _x 0 x^2 ((1)/(x)) Step 1: Understand the behavior of ((1)/(x)). For any x 0, the value of ((1)/(x)) is always between -1 and 1. -1 ((1)/(x)) 1 Step 2: Apply the Squeeze Theorem. Multiply the inequality by x^2. Since x^2 0, the direction of the inequalities remains the same. -x^2 x^2 ((1)/(x)) x^2 Step 3: Evaluate the limits of the bounding functions as x 0. _x 0 (-x^2) = -(0)^2 = 0 _x 0 (x^2) = (0)^2 = 0 Step 4: Conclude using the Squeeze Theorem. Since both bounding functions approach 0 as x 0, the limit of the middle function must also be 0. The limit is 0. 10. We need to evaluate the limit: _x -2 (|x+2|)/(x+2) Step 1: Define the absolute value function for x near -2. |x+2| = x+2 & if x+2 0 x -2 \\ -(x+2) & if x+2 < 0 x < -2 Step 2: Calculate the left-hand limit. As x -2^- (meaning x is slightly less than -2), x+2 < 0, so |x+2| = -(x+2). _x -2^- (-(x+2))/(x+2) = _x -2^- (-1) = -1 Step 3: Calculate the