Consider the piecewise functionf (x) =x + 1, if x < −21, if − 2 ≤ x ≤ 1x2, if x > 1.(i) Find limx→−2 f (x) if it exists.(ii) Show that f is continuous at x = 1.(iii) Sketch the graph of f (x)

(i) To find the limit of f(x) as x approaches -2, we need to consider the function definition for the interval that includes -2. According to the given piecewise function, f(x) = 1 when -2 ≤ x ≤ 1. Therefore, lim(x→-2) f(x) = 1.

(ii) To show that f is continuous at x = 1, we need to show that the left-hand limit, the right-hand limit, and the function value at x = 1 are all equal.

The left-hand limit as x approaches 1 is given by the function definition for -2 ≤ x ≤ 1, which is 1.

The right-hand limit as x approaches 1 is given by the function definition for x > 1, which is 1^2 = 1.

The function value at x = 1 is also 1 (from the definition for -2 ≤ x ≤ 1).

Since all these values are equal, f is continuous at x = 1.

(iii) To sketch the graph of f(x), we need to plot the three different function definitions on their respective intervals.

For x < -2, the graph is a straight line with slope 1 and y-intercept 1.

For -2 ≤ x ≤ 1, the graph is a horizontal line at y = 1.

For x > 1, the graph is a parabola opening upwards with vertex at (0,0).

The graph will show a sharp turn at x = -2 and a sharp turn at x = 1, reflecting the different function definitions at these points.