Find the indicated limits. If the limit does not exist, so state, or use the symbol + ∞ or - ∞.
f(x) = { 2 - x if x ≤ 3
{ -1 + 3x - x^2 if x > 3
a) lim 3+ f(x)
x->3
b) lim 3- f(x)
x->3
c) lim f(x)
x->3
d) lim ∞ f(x)
x->3
e) lim -∞ f(x)
x->3
To find the indicated limits, we will evaluate the function f(x) for different values of x approaching 3.
a) lim 3+ f(x) as x approaches 3 from the right:
For values of x greater than 3, the function is given by -1 + 3x - x^2. Let's evaluate this expression as x approaches 3 from the right.
lim 3+ f(x) = lim (x->3+) (-1 + 3x - x^2)
= (-1 + 3(3) - 3^2)
= (-1 + 9 - 9)
= -1
b) lim 3- f(x) as x approaches 3 from the left:
For values of x less than or equal to 3, the function is given by 2 - x. Let's evaluate this expression as x approaches 3 from the left.
lim 3- f(x) = lim (x->3-) (2 - x)
= (2 - 3)
= -1
c) lim f(x) as x approaches 3:
To find the limit at x = 3, we need to evaluate both the left-hand limit and the right-hand limit. Since the left-hand limit and the right-hand limit are both equal to -1, the overall limit is -1.
lim f(x) = -1
d) lim ∞ f(x) as x approaches 3:
This limit represents the behavior of the function as x approaches positive infinity. Since x approaches 3, not infinity, this limit is undefined.
e) lim -∞ f(x) as x approaches 3:
This limit represents the behavior of the function as x approaches negative infinity. Since x approaches 3, not negative infinity, this limit is also undefined.
Summary:
a) lim 3+ f(x) = -1
b) lim 3- f(x) = -1
c) lim f(x) = -1
d) lim ∞ f(x) = undefined
e) lim -∞ f(x) = undefined