Piecewise and Limit (Function) Question in Calculus?

Question

I need help on a piecewise function:

f(x) = { x if - 5
{ (x+2)^2 if - 3 < x
{ -x + 4 if x > 0


1. Which of the following can be conclude about limit x --> 0 f(x)

a) The limit does not exist since the right- sided and left- sided limits approaching x = 0 differ.

b) The limit does not exist because the function is not defined at x = 0.

c) The limit exists, and it’s equal to four.

d) The left-sided limit exists, but the right- sided limit does not. Thus, the limit does not exist.


2. What type of discontinuity occurs at x = -3?

a) No discontinuity exists at x=3.

b) Jump discontinuity.

c) Infinite discontinuity.

d) Removable discontinuity.


3. Determine the limit x --> infinity f(x) = ???


Thanks in advance...

Answer 1

1. near 0, the limit from above and the limit from below both = 4, so the limit is 4.

2. near -3, the limit from above = 1, from below = -5. since both limits exist and are different, this is a jump discontinuity.

3. as x gets very large, only the third branch applies, and f(x) gets very small. the limit is - infinity.

Answer 2

for the limit to exist the left hand and right hand limits must exist and they must be the same thing. You can ignore the first stuff (f(x) = {x if -5 0 the limit is -0 + 4 = 4. Since they're the same, the limit exists and it's value is 4. The answer is c.

Answer 3

1c, left and right side limits exists and are equal. it is not needed for f to be defined in x=0 ( although it is ) for the exisitance of the limit.

2b jump is not removable

3 that limit doesnt exist, because f is not bounded.