Calculus AB Limits Flashcards

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.