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 indicated one-sided limits, we need to evaluate the function f(x) separately for x approaching 0 from the right (x → 0^+) and from the left (x → 0^-).
For the function f(x) = [2x if x is less than 0] and [x^2 if x is greater than or equal to 0], let's evaluate each limit:
1. For the limit as x approaches 0 from the right (x → 0^+):
To find the limit as x approaches 0 from the right, we need to consider the expression [x^2 if x is greater than or equal to 0]. Since x → 0^+ means x is approaching 0 from the positive side, this indicates that x is greater than 0.
So the expression becomes x^2. Plugging in x = 0 into the expression x^2, we get:
lim f(x) as x → 0^+ = lim x^2 as x → 0^+ = 0^2 = 0
2. For the limit as x approaches 0 from the left (x → 0^-):
To find the limit as x approaches 0 from the left, we need to consider the expression [2x if x is less than 0]. Since x → 0^- means x is approaching 0 from the negative side, this indicates that x is less than 0.
So the expression becomes 2x. Plugging in x = 0 into the expression 2x, we get:
lim f(x) as x → 0^- = lim 2x as x → 0^- = 2(0) = 0
Thus, we can conclude that lim f(x) as x → 0^+ = 0 and lim f(x) as x → 0^- = 0, assuming the function is defined as described.
To find the indicated one-sided limit, we need to evaluate the function as x approaches the given value from the right (x->0^+) and from the left (x->0^-).
1. For the limit as x approaches 0 from the right (x->0^+):
lim f(x) = lim [2x] = lim 2x
x->0^+ x->0^+ x->0^+
Since this is a linear function, we can directly substitute the value x=0 into the expression:
lim f(x) = lim 2(0) = lim 0
x->0^+ x->0^+ x->0^+
Therefore, the limit as x approaches 0 from the right is 0.
2. For the limit as x approaches 0 from the left (x->0^-):
lim f(x) = lim [x^2] = lim x^2
x->0^- x->0^- x->0^-
Again, we can substitute x=0 directly into the expression:
lim f(x) = lim (0)^2 = lim 0
x->0^- x->0^- x->0^-
Therefore, the limit as x approaches 0 from the left is 0.
In conclusion:
- The limit as x approaches 0 from the right is 0.
- The limit as x approaches 0 from the left is also 0.