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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.
What are the differences between a limit and the value of a function at a point?
Limit: Value the function *approaches*. | Function Value: Value the function *is* at that point.
What are the differences between left-hand and right-hand limits?
Left-Hand: Approaching from the left. | Right-Hand: Approaching from the right.
What are the differences between a hole and a vertical asymptote on a graph?
Hole: Function undefined, limit may exist. | Vertical Asymptote: Function unbounded, limit DNE.
What are the differences between a jump discontinuity and a removable discontinuity?
Jump: Left and right limits differ. | Removable: Limit exists, but doesn't equal the function value or function not defined.
Compare estimating limits graphically vs. algebraically.
Graphically: Visual estimation, potential for inaccuracy. | Algebraically: Precise calculation, requires function definition.
Compare the limit of a continuous function vs. a discontinuous function.
Continuous: Limit often equals the function value. | Discontinuous: Limit may or may not exist; requires careful examination.
What are the differences between a one-sided limit existing and the overall limit existing?
One-sided: Function approaches a value from one direction. | Overall: Function approaches the same value from both directions.
Compare the behavior of a function near a vertical asymptote and near a hole.
Vertical Asymptote: Function values tend to infinity (or negative infinity). | Hole: Function is undefined, but values nearby are finite.
Compare the limit of sin(x) as x approaches 0 and sin(1/x) as x approaches 0.
sin(x): Limit is 0. | sin(1/x): Limit DNE due to oscillation.
Compare the limit of a polynomial function and a rational function.
Polynomial: Limit always exists and is easily found by direct substitution. | Rational: Limit may or may not exist; check for asymptotes and holes.
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|).