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
a) lim 3+ f(x)
x->3
To find the limit as x approaches 3 from the right, we look at the function for x > 3: -1 + 3x - x^2.
So, lim 3+ f(x) = -1 + 3(3) - (3)^2 = -1 + 9 - 9 = -1.
b) lim 3- f(x)
x->3
To find the limit as x approaches 3 from the left, we look at the function for x ≤ 3: 2 - x.
So, lim 3- f(x) = 2 - 3 = -1.
c) lim f(x)
x->3
Since the limits from both the left and the right are equal, the limit at x = 3 exists and is equal to -1.
d) lim ∞ f(x)
x->∞
As x goes to positive infinity, the second part of the function becomes dominant: -1 + 3x - x^2. Since the leading term is x^2 with a negative coefficient, the function will go to negative infinity as x approaches positive infinity:
lim ∞ f(x) = -∞
e) lim -∞ f(x)
x->-∞
As x goes to negative infinity, the first part of the function becomes dominant: 2 - x. Since the coefficient of x is negative, the function will go to positive infinity as x approaches negative infinity:
lim -∞ f(x) = ∞
a) To find lim 3+ f(x) as x approaches 3, we need to evaluate the limit as x approaches 3 from the right side of 3. In this case, since x is approaching 3 from the right side, we are looking at the function -1 + 3x - x^2.
So, we substitute x = 3 into the function:
lim 3+ f(x) = lim 3+ (-1 + 3x - x^2)
= -1 + 3(3) - (3)^2
= -1 + 9 - 9
= -1
Therefore, lim 3+ f(x) = -1.
b) To find lim 3- f(x) as x approaches 3, we need to evaluate the limit as x approaches 3 from the left side of 3. In this case, since x is approaching 3 from the left side, we are looking at the function 2 - x.
So, we substitute x = 3 into the function:
lim 3- f(x) = lim 3- (2 - x)
= 2 - (3)
= -1
Therefore, lim 3- f(x) = -1.
c) To find lim f(x) as x approaches 3, we need to find the limit of the function as x approaches 3 from both sides. Since lim 3+ f(x) = -1 and lim 3- f(x) = -1, and they are equal, we can conclude that lim f(x) = -1.
Therefore, lim f(x) = -1.
d) To find lim ∞ f(x) as x approaches 3, we need to evaluate the limit as x approaches infinity. In this case, the function -1 + 3x - x^2 is not defined for x = ∞. So, the limit does not exist.
Therefore, lim ∞ f(x) does not exist.
e) To find lim -∞ f(x) as x approaches 3, we need to evaluate the limit as x approaches negative infinity. In this case, the function -1 + 3x - x^2 is not defined for x = -∞. So, the limit does not exist.
Therefore, lim -∞ f(x) does not exist.
To find the indicated limits, we need to evaluate the function as x approaches 3 from the left (3-) and from the right (3+). We also need to check the limit as x approaches 3 and the limits as x approaches positive infinity (∞) and negative infinity (-∞). Let's go through each question step by step:
a) lim 3+ f(x) as x approaches 3 from the right:
To find this limit, we substitute x = 3 into the function for x > 3:
f(3) = -1 + 3(3) - (3^2) = -1 + 9 - 9 = -1
Therefore, the limit as x approaches 3 from the right is -1.
b) lim 3- f(x) as x approaches 3 from the left:
To find this limit, we substitute x = 3 into the function for x ≤ 3:
f(3) = 2 - 3 = -1
Therefore, the limit as x approaches 3 from the left is -1.
c) lim f(x) as x approaches 3:
Since both the left and right limits are equal to -1, the limit f(x) as x approaches 3 exists and is equal to -1.
d) lim ∞ f(x) as x approaches 3:
To find this limit, we need to consider the behavior of the function as x becomes very large. The function f(x) = -1 + 3x - x^2 approaches negative infinity as x approaches positive infinity. Therefore, the limit as x approaches 3 from positive infinity is -∞.
e) lim -∞ f(x) as x approaches 3:
To find this limit, we need to consider the behavior of the function as x becomes very small (negative). The function f(x) = 2 - x approaches positive infinity as x approaches negative infinity. Therefore, the limit as x approaches 3 from negative infinity is +∞.
In summary:
a) lim 3+ f(x) = -1
b) lim 3- f(x) = -1
c) lim f(x) = -1
d) lim ∞ f(x) = -∞
e) lim -∞ f(x) = +∞