Calculus AB Limits Flashcards
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Questions and Answers

What are limits?

  • The value a function approaches (correct)
  • The slope of a function
  • The minimum value of a function
  • The maximum value of a function

    • What does DNE stand for?

    Does Not Exist

    Unbounded limits result in DNE.

    True

    What does Lim f(x) x --> c^+ represent?

    <p>The limit as x approaches c from the right</p> Signup and view all the answers

    What does Lim f(x) x --> c^- represent?

    <p>The limit as x approaches c from the left</p> Signup and view all the answers

    The limit of f(x) x --> c exists when Lim f(x) x --> c^+ and Lim f(x) x --> c^- are the same.

    <p>True</p> Signup and view all the answers

    If a one-sided limit is unbounded, the limit exists.

    <p>False</p> Signup and view all the answers

    How can you estimate limits from a table?

    <p>Compare lower values from the left and higher values from the right.</p> Signup and view all the answers

    A function is undefined where the denominator equals what?

    <p>zero</p> Signup and view all the answers

    What is the sum property of limits?

    <p>Two limits added together equal the individual limits added together.</p> Signup and view all the answers

    What is the product property of limits?

    <p>Individual limits multiplied together equal the limit of the product.</p> Signup and view all the answers

    What is the Constant Multiple Rule?

    <p>d/dx[cf(x)] = cf'(x)</p> Signup and view all the answers

    What is the exponent limit rule?

    <p>The derivative raised to the exponent.</p> Signup and view all the answers

    What should you do with piecewise function limits if DNE occurs?

    <p>Break it up into left and right limits.</p> Signup and view all the answers

    What is 'f o g'?

    <p>f(g(x))</p> Signup and view all the answers

    What should be done when the internal or external limit does not exist?

    <p>Use the right and left limits to find the overall limit.</p> Signup and view all the answers

    What is Direct Substitution?

    <p>Evaluating a function by plugging a given value into the function.</p> Signup and view all the answers

    When evaluating a limit, the denominator can equal zero.

    <p>False</p> Signup and view all the answers

    How do you find limits of piecewise functions?

    <p>Use x and align it with the chart to find the correct function.</p> Signup and view all the answers

    What is the process for limits by factoring?

    <p>Factor out any crossing values, which become undefined, to get a 'new' function.</p> Signup and view all the answers

    What does (a+b)(a-b) equal?

    <p>a²-b²</p> Signup and view all the answers

    What does (√a+b)(√a-b) equal?

    <p>a-b²</p> Signup and view all the answers

    What is sin²x + cos²x equal to?

    <p>1</p> Signup and view all the answers

    What is cos 2x equal to?

    <p>cos²x - sin²x</p> Signup and view all the answers

    What is cos²x - sin²x equal to?

    <p>1 - 2sin²x</p> Signup and view all the answers

    What is 1 - 2sin²x equal to?

    <p>2cos²x - 1</p> Signup and view all the answers

    What is the Squeeze Theorem?

    <p>If f(x) ≤ g(x) ≤ h(x) and limx→a f(x) = limx→a h(x) = L, then limx→a g(x) = L.</p> Signup and view all the answers

    What is lim x->0 sin(x)/x equal to?

    <p>1</p> Signup and view all the answers

    What is lim -> 0 (1 - cos(x) / x) equal to?

    <p>0</p> Signup and view all the answers

    What is point discontinuity?

    <p>A point in the graph where the function does not exist, creating a removable hole.</p> Signup and view all the answers

    What is jump discontinuity?

    <p>A graph that has discontinuity where the function moves to a different y-value.</p> Signup and view all the answers

    What is asymptotic discontinuity?

    <p>A graph that approaches one or more asymptotes.</p> Signup and view all the answers

    F is continuous at x=c if lim f(x) x -> c = f(c).

    <p>True</p> Signup and view all the answers

    What is the average rate of change formula?

    <p>f(b) - f(a) / b - a</p> Signup and view all the answers

    What does the Mean Value Theorem state?

    <p>The instantaneous rate of change equals the mean rate of change somewhere in the interval.</p> Signup and view all the answers

    What does the Extreme Value Theorem state?

    <p>If f is continuous on [a,b], then f has an absolute maximum and minimum on [a,b].</p> Signup and view all the answers

    What are critical points?

    <p>Points in the domain of a function where f'=0 or f' does not exist.</p> Signup and view all the answers

    How do you find critical points?

    <p>Find the derivative, set to zero, and test using a number line.</p> Signup and view all the answers

    The function is decreasing when f'(x) < 0.

    <p>True</p> Signup and view all the answers

    What is relative maximum?

    <p>A point on the graph of a function where no other nearby points are higher.</p> Signup and view all the answers

    Study Notes

    Limits

    • Limits indicate the value that a function approaches as the variable approaches a certain point, denoted as ( \lim_{x \to c} f(x) ).
    • A limit can exist even if the function is undefined at that point.
    • DNE (Does Not Exist) occurs when a left-hand limit and right-hand limit yield different values.
    • Unbounded limits also result in DNE.

    One-sided Limits

    • ( \lim_{x \to c^+} f(x) ): Represents the limit as ( x ) approaches ( c ) from the right.
    • ( \lim_{x \to c^-} f(x) ): Represents the limit as ( x ) approaches ( c ) from the left.
    • If both one-sided limits are the same, the two-sided limit exists.

    Calculating Limits

    • Estimating limits using a table: Lower values indicate left limits, and higher values indicate right limits.
    • Direct substitution involves plugging the value directly into the function to evaluate limits.

    Properties of Limits

    • The Sum Property: ( \lim (f(x) + g(x)) = \lim f(x) + \lim g(x) )
    • The Product Property: ( \lim (f(x) \cdot g(x)) = \lim f(x) \cdot \lim g(x) )
    • Constant Multiple Rule: If ( c ) is a constant, then ( \frac{d}{dx}[c f(x)] = c f'(x) ).

    Special Limit Cases

    • The limit of ( \frac{\sin x}{x} ) as ( x \to 0 ) is 1.
    • The limit of ( \frac{1 - \cos x}{x} ) as ( x \to 0 ) is 0.

    Piecewise Functions

    • For limits involving piecewise functions, evaluate the limits separately for each side of the point of interest.
    • Factorization can help simplify limit evaluation; undefined values indicate crossings of the function.

    Types of Discontinuities

    • Point Discontinuity: Exists where the function has a hole in the graph; it's removable.
    • Jump Discontinuity: Function jumps to a different y-value, non-removable.
    • Asymptotic Discontinuity: Approaches an asymptote, resulting in non-removable discontinuity.

    Continuity and Critical Points

    • A function ( f ) is continuous at ( x = c ) if ( \lim_{x \to c} f(x) = f(c) ).
    • A critical point occurs where ( f' = 0 ) or ( f' ) is undefined, which may indicate local maxima or minima.

    Theorems

    • Mean Value Theorem: There exists at least one point where the instantaneous rate of change equals the average rate of change on an interval.
    • Extreme Value Theorem: A continuous function on a closed interval has both an absolute maximum and minimum.

    Additional Trigonometric Identities

    • ( \sin^2 x + \cos^2 x = 1 )
    • ( \cos(2x) = \cos^2 x - \sin^2 x )
    • Alternative forms include ( \cos^2 x - \sin^2 x = 1 - 2 \sin^2 x ) and ( 1 - 2 \sin^2 x = 2 \cos^2 x - 1 ).

    Finding Critical Points

    • To find critical points, take the derivative, set it to zero, and test the intervals using a number line.

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    Test your knowledge of limits in Calculus AB with these flashcards. Each card covers important concepts and terminology related to limits, including definitions and conditions. Perfect for reviewing key ideas before exams.

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