Find the indicated one-sided limit, if it exists
1.lim f(x) and lim f(x), where
x->0^+ x->0^-
2. f(x) = [2x if x is less than 0]
[x^2 if x is greater than or equal to 0]
To find the one-sided limit, we need to determine the limit of the function as x approaches the given value from the left (x->0^-) and from the right (x->0^+).
1. Let's find the one-sided limits for the given function:
a) x->0^-
To find the limit as x approaches 0 from the left, we evaluate the function as x approaches 0 but with values slightly less than 0.
For this particular function, which is not defined for x < 0, the limit as x approaches 0 from the left does not exist.
b) x->0^+
To find the limit as x approaches 0 from the right, we evaluate the function as x approaches 0 but with values slightly greater than 0.
Since there is no specific expression given for the function f(x) when x > 0, we are unable to determine the limit as x approaches 0 from the right. Therefore, it does not exist.
2. Let's find the one-sided limits for the second function:
a) x->0^-
To find the limit as x approaches 0 from the left, we evaluate the function as x approaches 0 but with x-values slightly less than 0.
Since the expression for f(x) when x < 0 is 2x, we substitute this into the limit expression:
lim (2x) as x->0^- = 2(0) = 0
b) x->0^+
To find the limit as x approaches 0 from the right, we evaluate the function as x approaches 0 but with x-values slightly greater than 0.
Since the expression for f(x) when x ≥ 0 is x^2, we substitute this into the limit expression:
lim (x^2) as x->0^+ = (0)^2 = 0
Therefore, the one-sided limits of the function f(x) = [2x if x is less than 0] and [x^2 if x is greater than or equal to 0] are:
- lim f(x) as x->0^- = 0
- lim f(x) as x->0^+ = 0