All Flashcards
What does a horizontal line on a graph indicate about its limit?
If a function approaches a horizontal line as x approaches a certain value, the limit at that value is the y-value of the horizontal line.
How do you interpret a hole in a graph when finding a limit?
A hole indicates that the function is not defined at that specific x-value, but the limit may still exist if the function approaches a specific y-value from both sides.
How does a steep slope on a graph relate to the existence of a limit?
A steep slope doesn't directly indicate whether a limit exists, but if the slope becomes infinitely steep (vertical asymptote), the limit likely does not exist.
How does the graph of a piecewise function help in evaluating limits?
It shows different function definitions for different intervals, requiring you to check one-sided limits at the points where the definition changes.
Given a graph, how can you tell if the limit as x approaches infinity exists?
Check if the function approaches a horizontal asymptote as x goes to infinity. If it does, that is the limit.
How can you use a graph to approximate the limit of a function as x approaches a specific value?
By visually inspecting the graph, trace the function from both the left and right sides towards the x-value of interest. The y-value the function approaches is the approximate limit.
What does a graph with many sharp corners suggest about the function's limit?
Sharp corners can suggest that the function may not be differentiable at those points, but it doesn't necessarily mean the limit doesn't exist. You still need to check one-sided limits.
How do you interpret the graph of f(x) = c (a constant function) when finding limits?
The limit of a constant function as x approaches any value is simply the constant value itself. The graph is a horizontal line.
How does the graph of f(x) = x help in understanding limits?
The limit of f(x) = x as x approaches 'a' is simply 'a'. The graph is a straight line passing through the origin with a slope of 1.
What does the graph of an absolute value function tell us about its limits?
The limit exists everywhere, but the derivative does not exist at the corner (e.g., at x=0 for |x|).
What is a limit?
The value a function approaches as the input approaches a specific point.
Define a left-hand limit.
The value a function approaches as x approaches a value from the left.
Define a right-hand limit.
The value a function approaches as x approaches a value from the right.
What does DNE mean in the context of limits?
Does Not Exist. The limit does not approach a single, finite value.
What is a vertical asymptote?
A vertical line that a function approaches but never touches; often indicates a limit DNE.
What is a jump discontinuity?
A point where the left-hand limit and right-hand limit exist but are not equal.
What is an oscillating function?
A function that fluctuates rapidly between two values as x approaches a certain point.
What is an unbounded function?
A function whose values increase or decrease without limit as x approaches a certain point.
What is the notation for the limit of f(x) as x approaches 'a'?
What does it mean for a function to be continuous at a point?
The limit exists at that point, the function is defined at that point, and the limit equals the function value.
What is the notation for the left-hand limit as x approaches a?
What is the notation for the right-hand limit as x approaches a?