Give the value of each statement. If the value does not exist, write "does not exist" or "undefined".
Evaluate the following from the graph given below.
Problem 1 :
lim_{ x→−1}^{-} f(x)
Solution :
Approaching -1 from left side, we get the value of y as 3. So,
lim _{x→-1}^{-} f(x) = 3
Problem 2 :
f(1)
Solution :
We see the filled circle at (1, 1). So, the value of f(1) is 1.
Problem 3 :
lim _{x→0} f(x)
Solution :
At exactly x approaches 0, the output is 0.
lim _{x→0} f(x) = 0.
Problem 4 :
lim _{x→2}+ f(x)
Solution :
Approaching 2 from right side, the output is 1.
So, lim _{x→2}^{+} f(x) = 1
Problem 5 :
f(-1)
Solution :
We see the filled circle at (-1, 1). So, the value of f(-1) is 1.
Problem 6 :
f(2)
Solution :
f(2) = does not exists.
Problem 7 :
lim _{x→−1}^{+} f(x)
Solution :
Approaching -1 from right side, we get the value of y as 1. So,
lim _{x→-1}^{+} f(x) = 1
Problem 8 :
lim _{x→1}^{-} f(x)
Solution :
Approaching 1 from left side, we get the value of y as -1. So,
lim_{ x→1}^{- } f(x) = -1
Problem 9 :
lim_{ x→2} f(x)
Solution :
Both left hand and right hand limits are not equal, the limit does not exists at x = 2.
lim _{x→2} f(x) = DNE
Evaluate the following from the graph given below.
Problem 1 :
lim _{x→-3} f(x)
Solution :
lim _{x}_{ →-3} f(x)
At x = -3, the curve touches the x-axis. So, the output is 0.
lim _{x →-3} f(x) = 0
Problem 2 :
f(1)
Solution :
The point f(1) does not pass through any points. So, the answer is does not exists.
Problem 3 :
lim _{x→1} f(x)
Solution :
Left hand limits are not equal. Then right hand limit is exists at x = 1.
lim _{x→1} f(x) = -5
Problem 4 :
lim _{x→-2}+ f(x)
Solution :
Approaching -2 from right side, we get the value of y as 4. So,
lim _{x→-2}^{+ } f(x) = 4
Problem 5 :
f(3)
Solution :
The curve passes through the point (3, -1).
f(3) = -1
Problem 6 :
lim _{x→-2}^{-}f(x)
Solution :
Approaching -2 from left side, we get the value of y as 1. So,
lim _{x →-2}^{-} f(x) = 1
Problem 7 :
lim _{x→-2 } f(x)
Solution :
Both left hand and right hand limits are not equal, the limit does not exists at x = -2.
lim _{x→-2} f(x) = DNE
Problem 8 :
f(-2)
Solution :
We see the filled circle at (-2, 3). So, the value of f(-2) is 3.
Problem 9 :
f(4)
Solution :
The curve passes through the point (4, 1), so the value of f(4) is 1.
Evaluate the following from the graph given below.
Problem 1 :
lim _{x→3}^{+} f(x)
Solution :
Approaching 3 from right side, we get the value of y as 1. So,
lim _{x→3}^{+} f(x) = 1
Problem 2 :
f(3)
Solution :
The curve does not pass through any points on the y-axis. So, the answer is does not exists.
Problem 3 :
lim _{x→0} f(x)
Solution :
At exactly x approaches 0, the output is 1.
lim _{x→0} f(x) = 1.
Problem 4 :
lim _{x→3} f(x)
Solution :
Both left hand and right hand limits are not equal, the limit does not exists at x = 3.
lim_{ x→3} f(x) = DNE
Problem 5 :
f(0)
Solution :
The curve is passing through (0, 2). So, the value of f(0) is 2.
Problem 6 :
lim _{x→3}^{-} f(x)
Solution :
Approaching 3 from right side, we get the value of y as -2. So,
lim _{x→3}^{-} f(x) = -2
Problem 7 :
lim _{x→0}^{+} f(x)
Solution :
While approaching the value 0 from left side, the output becomes 1.
So, lim _{x→0}+ f(x) = 1.
Problem 8 :
f(1)
Solution :
The curve is passing through (1, 0). So, the value of f(1) is 0.
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