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|).
How do you estimate a limit from a graph?
1. 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?
1. 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 $$ \lim_{x \to a} f(x) $$ from a graph?
Determine $$ \lim_{x \to a^-} f(x) $$ and $$ \lim_{x \to a^+} f(x) $$. 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 $$ \lim_{x \to a} f(x) $$?
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 $$ \lim_{x \to a} f(x) = f(a)$$.
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.
