All Flashcards
Define 'limit'.
The value that a function approaches as the input approaches some value.
What is a horizontal asymptote?
A y-value that the graph of a function approaches as x tends to +∞ or -∞.
Define 'end behavior'.
The behavior of a function as x approaches positive or negative infinity.
What is an oscillating function?
A function that repeatedly fluctuates between two or more values as x approaches infinity.
Define the Squeeze Theorem.
If g(x) ≤ f(x) ≤ h(x) for all x near c, and lim x→c g(x) = L = lim x→c h(x), then lim x→c f(x) = L.
What is meant by 'growth rate' in the context of functions?
The speed at which a function's output increases as its input increases.
Define a rational function.
A function that can be defined as a fraction where both the numerator and denominator are polynomials.
What is the degree of a polynomial?
The highest power of the variable in the polynomial.
Define 'leading coefficient'.
The coefficient of the term with the highest power in a polynomial.
What does DNE stand for in the context of limits?
Does Not Exist
What are the differences between limits at infinity and limits at a finite number?
Limits at infinity: x approaches ∞ or -∞, examining end behavior. Limits at a finite number: x approaches a specific value, examining local behavior.
What are the differences between horizontal and vertical asymptotes?
Horizontal: y-value function approaches as x approaches infinity. Vertical: x-value function approaches infinity.
Compare the growth rates of polynomial and exponential functions.
Polynomial: Grows at a polynomial rate (e.g., , ). Exponential: Grows at an exponential rate (e.g., , ). Exponential growth is much faster.
Compare the behavior of and as x approaches infinity.
Both approach 0. can be positive or negative depending on the sign of x, while is always positive.
Contrast the limits of and as x approaches infinity.
: Limit DNE (oscillates). : Limit is 0 (Squeeze Theorem).
Compare the end behavior of and .
: As x approaches ±∞, approaches +∞. : As x approaches +∞, approaches +∞. As x approaches -∞, approaches -∞.
Compare horizontal asymptotes with removable discontinuities.
Horizontal Asymptotes: Describe the end behavior of a function. Removable Discontinuities: Points where the function is not defined but could be redefined to be continuous.
Compare the limits of and as x approaches infinity.
: Approaches infinity, but very slowly. : Approaches infinity very quickly.
What is the difference between an infinite limit and a limit at infinity?
Infinite Limit: The function approaches infinity as x approaches a finite value. Limit at Infinity: The function approaches a finite value or infinity as x approaches infinity.
Compare the limits of and as x approaches infinity.
: Approaches 1. : Approaches 1.
If a graph approaches y = 2 as x goes to infinity, what does this mean?
The function has a horizontal asymptote at y = 2, and .
If a graph has a horizontal asymptote at y = 0, what does this imply about the function's behavior at infinity?
The function's y-values approach 0 as x goes to positive or negative infinity.
How can you identify a horizontal asymptote on a graph?
Look for a horizontal line that the graph approaches but does not cross (at least at the extreme ends).
How can you identify a vertical asymptote on a graph?
Look for a vertical line where the function approaches infinity or negative infinity.
What does a graph with no horizontal asymptote indicate?
The function's y-values do not approach a finite limit as x goes to positive or negative infinity.
What does the graph of f(x) = e^x look like as x approaches negative infinity?
The graph approaches the x-axis (y=0) from above.
What does the graph of f(x) = ln(x) look like as x approaches infinity?
The graph increases without bound, but at a decreasing rate.
How to interpret the graph of as x approaches infinity?
The graph oscillates with decreasing amplitude, approaching y = 0.
What does it mean if a graph crosses its horizontal asymptote?
The function's value equals the value of the horizontal asymptote at that point. This is possible, but the function must still approach the asymptote as x goes to infinity.
What can you infer if a function has a horizontal asymptote at y=L?
and