Questions and Answers
What are limits?
What does DNE stand for?
Does Not Exist
Unbounded limits result in DNE.
True
What does Lim f(x) x --> c^+ represent?
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What does Lim f(x) x --> c^- represent?
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The limit of f(x) x --> c exists when Lim f(x) x --> c^+ and Lim f(x) x --> c^- are the same.
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If a one-sided limit is unbounded, the limit exists.
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How can you estimate limits from a table?
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A function is undefined where the denominator equals what?
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What is the sum property of limits?
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What is the product property of limits?
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What is the Constant Multiple Rule?
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What is the exponent limit rule?
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What should you do with piecewise function limits if DNE occurs?
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What is 'f o g'?
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What should be done when the internal or external limit does not exist?
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What is Direct Substitution?
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When evaluating a limit, the denominator can equal zero.
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How do you find limits of piecewise functions?
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What is the process for limits by factoring?
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What does (a+b)(a-b) equal?
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What does (√a+b)(√a-b) equal?
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What is sin²x + cos²x equal to?
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What is cos 2x equal to?
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What is cos²x - sin²x equal to?
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What is 1 - 2sin²x equal to?
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What is the Squeeze Theorem?
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What is lim x->0 sin(x)/x equal to?
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What is lim -> 0 (1 - cos(x) / x) equal to?
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What is point discontinuity?
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What is jump discontinuity?
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What is asymptotic discontinuity?
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F is continuous at x=c if lim f(x) x -> c = f(c).
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What is the average rate of change formula?
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What does the Mean Value Theorem state?
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What does the Extreme Value Theorem state?
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What are critical points?
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How do you find critical points?
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The function is decreasing when f'(x) < 0.
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What is relative maximum?
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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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