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|).
How do you estimate a limit from a graph?
- Visualize the point. 2. Trace along the graph from both sides. 3. Check if the one-sided limits match.
How do you determine if a limit DNE from a graph?
- Check for vertical asymptotes. 2. Check for jump discontinuities. 3. Check for wild oscillations.
Given a graph, how do you find the left-hand limit at x=a?
Trace the graph from the left side of x=a. Determine the y-value the function approaches as x gets closer to a from the left.
Given a graph, how do you find the right-hand limit at x=a?
Trace the graph from the right side of x=a. Determine the y-value the function approaches as x gets closer to a from the right.
How do you evaluate from a graph?
Determine and . If they are equal, that is the limit. Otherwise, the limit DNE.
How do you handle a graph with a hole at x=a when finding ?
The limit can still exist even with a hole. Focus on what y-value the function approaches as x approaches 'a' from both sides, not the value at x=a.
How do you determine if a function is continuous at x=a from its graph?
Check if the limit exists at x=a, if f(a) is defined, and if .
How do you identify a jump discontinuity on a graph?
Look for a point where the graph 'jumps' from one y-value to another. The left and right limits will be different at this point.
How do you identify a vertical asymptote on a graph?
Look for a vertical line where the function approaches infinity (or negative infinity) as x approaches that line.
How do you deal with oscillations when estimating limits from a graph?
If the function oscillates wildly near a point, the limit likely does not exist. The function does not approach a single, finite value.
Explain the concept of a limit.
A limit describes the value that a function approaches as the input approaches some value. It focuses on the behavior near a point, not necessarily at the point itself.
Explain one-sided limits.
One-sided limits examine the behavior of a function as it approaches a value from either the left (left-hand limit) or the right (right-hand limit).
When does a limit not exist?
A limit does not exist if the function is unbounded, oscillates wildly, or if the left-hand limit and right-hand limit are not equal.
How can graphs be used to estimate limits?
By observing the y-value that the function approaches as x approaches a specific value. Trace the curve from both sides to see if they converge to the same y-value.
What does a jump discontinuity indicate about limits?
A jump discontinuity indicates that the left-hand limit and the right-hand limit are different, therefore the limit at that point does not exist.
What does a vertical asymptote indicate about limits?
A vertical asymptote indicates that the function is unbounded as x approaches a certain value, meaning the limit does not exist at that point.
What is the relationship between continuity and limits?
For a function to be continuous at a point, the limit must exist at that point, the function must be defined at that point, and the limit must equal the function value.
Explain why scale is important when estimating limits from graphs.
The scale of a graph can distort the appearance of the function, potentially hiding important details or exaggerating certain behaviors, leading to incorrect limit estimations.
Why is it important to check one-sided limits?
Checking one-sided limits is crucial because the overall limit exists only if both the left-hand limit and the right-hand limit exist and are equal.
Explain the concept of oscillation in the context of limits.
Oscillation refers to a function fluctuating rapidly between two values as x approaches a specific point. If the oscillations become infinitely rapid, the limit does not exist.