Question

Sketch a function p(x) on the domain (-5, 5] that contains the entire list of

features below.
a) lim p(x) = 00
2. Identify any and all
discontinuities by the type and
the x-value.
b) lim p(x) = -00
c) lim p(x) = 1
d) p(0) = -2
e) lim p(x) = 2
X-3
f) lim p(x) = -1
X-3
g) lim p(x) = DNE
h) p(-3) = -1
i) p(-5) = p(5) = 0

Answer 1

Sketch a function p(x) that includes a jump discontinuity at x = -3, vertical asymptotes, and passes through specific points, ensuring limits match given conditions.

The task is to sketch a function p(x) satisfying several conditions on the domain (-5, 5]. Here is a possible step-by-step explanation to match the given requirements:

1. At x = -5 and x = 5, set the function value to 0 to satisfy the condition i) p(-5) = p(5) = 0.
2. Create an asymptote at x = 2 to satisfy conditions e) and f) as lim p(x) approaches 2 for x approaching 3 from the left, and -1 for x approaching 3 from the right.
3. Ensure p(-3) = -1 to satisfy condition h) and introduce a discontinuity of the type 'jump discontinuity' at x = -3 with a limit that does not exist, satisfying condition g).
4. Ensure p(0) = -2, as specified in condition d).
5. To satisfy conditions a), b), and c), create vertical asymptotes and horizontal asymptotes where necessary such that the limits approach the specified infinity or value.

p(x) will have a jump discontinuity at x = -3 and vertical asymptotes where the limits approach infinity. It will pass through the points (-5,0), (-3,-1), and (0,-2) and will have the appropriate behavior at x = 3 to satisfy the limits from both sides.