All Flashcards
What are the differences between direct substitution and algebraic manipulation for finding limits?
Direct Substitution: Plug in the value directly. | Algebraic Manipulation: Simplify the expression before plugging in the value (factoring, conjugates, etc.).
What are the differences between using a graph and a table to find limits?
Graph: Visual representation of the function's behavior. | Table: Numerical representation of the function's values.
What are the differences between L'Hôpital's Rule and the Squeeze Theorem?
L'Hôpital's Rule: Used for indeterminate forms by taking derivatives. | Squeeze Theorem: Used to find limits by bounding a function between two others.
What are the differences between one-sided limits and two-sided limits?
One-sided limits: Approach from either the left or the right. | Two-sided limits: Approach from both sides, and both must agree for the limit to exist.
What are the differences between a limit existing and a function being continuous at a point?
Limit exists: The function approaches a value. | Continuous: The limit exists, the function is defined, and they are equal.
What are the differences between removable and non-removable discontinuities?
Removable: Can be 'fixed' by redefining the function at a single point. | Non-removable: Cannot be fixed (e.g., jump, asymptote).
What are the differences between finding limits at finite values and finding limits at infinity?
Finite values: Focus on the function's behavior near a specific point. | Infinity: Focus on the function's end behavior as x grows without bound.
What are the differences between using limit laws and algebraic manipulation?
Limit Laws: Apply basic properties of limits (sum, product, etc.). | Algebraic Manipulation: Transform the expression to simplify it before applying limit laws.
What are the differences between indeterminate forms 0/0 and ∞/∞?
0/0: Both numerator and denominator approach zero. | ∞/∞: Both numerator and denominator approach infinity.
What are the differences between using conjugates and factoring when simplifying limits?
Conjugates: Used for expressions with radicals. | Factoring: Used for polynomial expressions.
Explain how to find a limit from a graph.
Look at the y-value the graph approaches as x approaches a certain value. The limit is what y is heading toward, not necessarily the value at that point.
Explain how to estimate a limit from a table.
Analyze the trend of y-values as x-values get closer to a specific point. Look for patterns and the value y is approaching.
Explain how algebraic properties are used to determine limits.
Use limit laws to break down complex functions into simpler ones. Limits work well with addition, subtraction, multiplication, and division. Consider composite functions.
Explain how algebraic manipulation is used to determine limits.
Manipulate the function to eliminate indeterminate forms (like 0/0). Use conjugates, L'Hôpital's rule, or simplifying rational functions.
When should you use a visual representation to find a limit?
When you are given a graph. Scan the graph to see what y-value the function approaches as x approaches a certain value.
When should you use a table to find a limit?
When you are given a table of x and y values. Look for patterns and see where the y-values are heading as the x-values approach a specific point.
When should you use algebraic properties to find a limit?
When you see basic limit theorems (sums, differences, products, quotients) or composite functions. Break down the problem into simpler parts.
When should you use algebraic manipulation to find a limit?
When you have complex functions that need simplification. Think conjugates, L'Hôpital's Rule, or simplifying rational functions.
Explain the concept of the Squeeze Theorem.
If a function is bounded above and below by two other functions that have the same limit at a certain point, then the function in the middle must also have the same limit at that point.
Explain why it's important to check for indeterminate forms before applying L'Hopital's Rule.
L'Hopital's Rule can only be applied to indeterminate forms (0/0 or ∞/∞). Applying it to other forms will lead to incorrect results.
How does a hole in a graph relate to limits?
A hole indicates that the function is not defined at that x-value, but the limit may still exist if the function approaches a specific y-value.
How does a vertical asymptote on a graph relate to limits?
A vertical asymptote indicates that the limit approaches infinity (or negative infinity) as x approaches a certain value from one or both sides.
What does it mean if a graph oscillates rapidly near a point when considering limits?
It suggests that the limit does not exist because the function does not approach a single value.
How can you identify one-sided limits from a graph?
Examine the behavior of the function as x approaches the value from the left and from the right separately.
How does the graph of a function indicate continuity at a point?
The graph must not have any breaks, holes, or jumps at that point. You should be able to draw the graph through that point without lifting your pen.
How can a graph help visualize the Squeeze Theorem?
You can see the function being 'squeezed' between two other functions that converge to the same limit.
What does the graph of y = sin(x)/x tell us about its limit as x approaches 0?
The graph visually shows that as x approaches 0, the function approaches 1, even though the function is undefined at x = 0.
How does the slope of a tangent line on a graph relate to limits?
The slope of the tangent line at a point is the limit of the difference quotient as h approaches 0, which represents the derivative at that point.
How can you use a graph to determine infinite limits?
Look for vertical asymptotes. If the function approaches infinity (or negative infinity) as x approaches a certain value, then the limit is infinite.
How does a jump discontinuity affect the limit of a function?
The limit does not exist at a jump discontinuity because the left-hand limit and the right-hand limit are not equal.