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Questions and Answers
In a Cartesian plane, where are negative x values (x < 0) located?
In a Cartesian plane, where are negative x values (x < 0) located?
- At the top
- On the left-hand side (correct)
- On the right-hand side
- At the bottom
When determining limits on a Cartesian plane, it is unnecessary to observe the behavior to the left and right of a chosen point.
When determining limits on a Cartesian plane, it is unnecessary to observe the behavior to the left and right of a chosen point.
False (B)
What term is used to describe a point on the x-axis used as a reference when navigating functions on a Cartesian plane?
What term is used to describe a point on the x-axis used as a reference when navigating functions on a Cartesian plane?
a
For a piecewise function, the point 'a' could be the __________ for function intervals, where one sub-function ends and another begins.
For a piecewise function, the point 'a' could be the __________ for function intervals, where one sub-function ends and another begins.
Associate each function definition with its corresponding condition within the piecewise function:
Associate each function definition with its corresponding condition within the piecewise function:
In the analogy presented, what does 'x = a' represent?
In the analogy presented, what does 'x = a' represent?
When 'crossing the road' at x = a, it is only necessary to check for oncoming traffic from one direction.
When 'crossing the road' at x = a, it is only necessary to check for oncoming traffic from one direction.
What must be evaluated regarding the 'cars' (or function behavior) on a graph to safely 'cross the road' at x = a?
What must be evaluated regarding the 'cars' (or function behavior) on a graph to safely 'cross the road' at x = a?
In terms of limits, safely 'crossing the road' means avoiding the same ______ coordinate when the approaching functions reach x = a.
In terms of limits, safely 'crossing the road' means avoiding the same ______ coordinate when the approaching functions reach x = a.
Match the piecewise function notation to its description in the 'crossing the road' analogy.
Match the piecewise function notation to its description in the 'crossing the road' analogy.
Under which condition(s) in the piecewise function does the road meet?
Under which condition(s) in the piecewise function does the road meet?
In the example piecewise function y = 1 if x < 1, and y = x if x ≥ 1, the function approaches different values from the left and right sides of x = 1.
In the example piecewise function y = 1 if x < 1, and y = x if x ≥ 1, the function approaches different values from the left and right sides of x = 1.
What must you determine about f(x) to safely cross the road on a function?
What must you determine about f(x) to safely cross the road on a function?
To properly analyze a piecewise function, you must find out the __________ limits of f(x).
To properly analyze a piecewise function, you must find out the __________ limits of f(x).
Correspond each limit notation with its accurate clarification:
Correspond each limit notation with its accurate clarification:
In the given example, lim x→1- is the limit that approaches from which side of the x-axis?
In the given example, lim x→1- is the limit that approaches from which side of the x-axis?
Using direct substitution, the limit as x approaches 1 from the right is equal to the function value at x = 2.
Using direct substitution, the limit as x approaches 1 from the right is equal to the function value at x = 2.
What condition is satisfied when both left and right limits are equal and identical to the function's value at a certain point?
What condition is satisfied when both left and right limits are equal and identical to the function's value at a certain point?
If lim x→1- f(x) = lim x→1+ f(x) = f(1), it means the road is __________ and cars can move smoothly.
If lim x→1- f(x) = lim x→1+ f(x) = f(1), it means the road is __________ and cars can move smoothly.
Organize each statement to whether it applies for continuity:
Organize each statement to whether it applies for continuity:
Flashcards
Cartesian Plane
Cartesian Plane
The basic plane where x is negative on the left (x < 0) and positive on the right (x > 0).
Reference Point 'a'
Reference Point 'a'
The point on the x-axis used when navigating functions on a Cartesian plane to determine limits and piecewise functions.
Piecewise Function
Piecewise Function
A function defined by multiple sub-functions, each applying to a certain interval of the input.
One-Sided Limits
One-Sided Limits
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Left-Hand Limit
Left-Hand Limit
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Right-Hand Limit
Right-Hand Limit
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Continuous Function
Continuous Function
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Continuity Condition
Continuity Condition
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Study Notes
- Notes on limits and piecewise functions are below
Cartesian Plane Points
- Cartesian plane basics: x is negative (x < 0) on the left, positive (x > 0) on the right.
- Use points on the x-axis as references when working with functions to determine limits and piecewise functions.
- Refer to a point on the x-axis as "a".
- Check what happens to the left and right of "a" before calculations.
- Point "a" can be the endpoint for piecewise function intervals.
- Piecewise functions have sub-functions, one ends, and another begins at "a".
- A piecewise function can be represented as
f(x) = g(x), if x < a
h(x), if x ≥ a
Crossing a Road
- Imagine looking at a road with a bend on a Cartesian plane.
- You are on the x-axis at x = a, looking north at the bend.
- To cross safely, look at cars coming from the left (x < a) and right (x ≥ a).
- Observe how cars behave as they approach x = a to avoid being at the same y coordinate when they reach that point.
Piecewise Functions
- Road as a piecewise function:
f(x) = g(x), if x < a
h(x), if x ≥ a
- The two parts of the road meet where x = a and y = g(a) = h(a).
- Piecewise function of the road:
y = f(x) = 1, if x < 1
x, if x ≥ 1
- To cross the road safely at x = a = 1, it's important to know what happens as the function approaches x = 1, and knowing the limit as f(x) approaches x = 1.
- Since f(x) is a piecewise function with intervals ending at x = 1, the road behaves differently to the left and right.
- Determine the one-sided (directional) limits of f(x):
lim f(x) and lim f(x)
x-1- x-1+
One-Sided Limits
- Left limit: limx→1-
- Approaching from the negative side of the x-axis (like the blue car).
- Notation: limx→1- f(x).
- This means checking the region where x < 1.
- In this region, the sub-function of f(x) is y = 1.
- Therefore, limx→1- f(x) = limx→1- 1 = 1.
- The limit of a constant is the constant itself.
- Right limit: limx→1+
- Approaching from the positive side of the x-axis (like the red car).
- Notation: limx→1+ f(x).
- This means checking the region where x > 1.
- In this region, the sub-function of f(x) is y = x.
- Therefore, limx→1+ f(x) = limx→1+ x = 1 using direct substitution.
Continuity
- From one-sided limits: limx→1- f(x) = 1 = limx→1+ f(x).
- Since the left and right limits are equal, the limit exists at that point: limx→1 f(x) = 1.
- f(1) = 1, as defined from the right side of the piecewise function.
- Therefore, limx→1 f(x) = 1 = f(1).
- f(1) = 1, defined value at x = 1.
- limx→1 f(x) = 1, the limit at x = 1 exists.
- limx→1 f(x) = f(1), the limit equals the function value.
- The road is continuous with cars moving freely, exercise caution when crossing.
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