All Flashcards
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.
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 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.