Another problem with limits?

Question

Find the limit (if it exists)

f(x) =　{ x^2 + 1, x less than or greater than 0
　　　　{ 2x + 3 , x > 0

a) lim = f(x)
　x > 0 -

b) lim = f(x)
　x > 0 +

c) lim = f(x)
　x > 0

a) x -> 0 -

x^2 + 3 = (0)^2 + 1 = 1


b) x -> 0 +

2x+3 = 2(0) + 3 = 3

The limit does not exist for both of these right?

c)


x^2 + 1 = (0)^2 + 1 = 1

Did I pick the right one?

oops. its less than or equal to for part A

≤

So for the answers, it should look like..

A) DNE
B) DNE
C) 1

?

ok.

A) 1
B) 3
C) DNE

Answer 1

Your answers to a) and b) are correct. Your answer to c) is: a limit as x --> 0 does not exist because the left-hand and right-hand limits are not equal.

Answer 2

Well, you know that technically, the first function is going to be to the left side of 0 and the second function is going to be to the right side of 0.

a) lim (x > 0-) means when it's approaching from the left. So solve the first equation, since it's the only one of the two that includes values of x that are less than 0. Solve it for x=0, and see if you get an answer:
x^2 + 1 = f(x)
0 + 1 = f(x)
1 = f(x) that's your limit from the left (lim (x > 0-))

b) lim (x > 0+) means when it's approaching from the right. So solve the second equation, since it is the only one of the two that includes values greater than 0. Solve it for x=0, even though it's only meant for values greater than 0, because it is the only one of the two eqn's that has ANYTHING to the right of 0. Remember, it's all about from where it approaches:
lim (x > 0+) = f(x)
2x + 3 = f(x)
0 + 3 = f(x)
3 = f(x) that's your limit from the right (f(x) = 3)

c) DNE, because the two limits that you found earlier are not equal.

Answer 3

f(x) = x² + 1 for x less than or GREATER THAN 0? I assume you mean x ≤ 0, since the other condition is x > 0.

a) lim (x→0-) = 1
b) lim (x→0+) = 3
c) since a) and b) are not the same, lim (x→0) does not exist.

Answer 4

Both limits exist (as you've correctly demonstrated in parts (a) and (b)), but they are not equal. Therefore, there is no general limit at 0. This is the definition of a discontinuity in the function.