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Chapter 2 | Limits
CHAPTER 2 REVIEW
average velocity KEY TERMS
the change in an object’s position divided by the length of a time period; the average velocity of an object over a time interval [ t , a ] (if t < a or [ a , t ] if t > a ) , with a position given by s ( t ), that is v ave = s ( t )− s ( a ) t − a the limit law lim x → a cf ( x ) = c · lim x → a f ( x ) = cL A function f ( x ) is continuous at a point a if and only if the following three conditions are satisfied: (1) f ( a ) is defined, (2) lim x → a f ( x ) exists, and (3) lim x → a f ( x ) = f ( a ) A function is continuous from the left at b if lim x → b − f ( x ) = f ( b ) A function is continuous from the right at a if lim x → a + f ( x ) = f ( a ) a function that can be traced with a pencil without lifting the pencil; a function is continuous over an open interval if it is continuous at every point in the interval; a function f ( x ) is continuous over a closed interval of the form ⎡ ⎣ a , b ⎤ ⎦ if it is continuous at every point in ( a , b ), and it is continuous from the right at a and from the left at b the limit law lim x → a ⎛ ⎝ f ( x )− g ( x ) ⎞ ⎠ = lim x → a f ( x )− lim x → a g ( x ) = L − M the field of calculus concerned with the study of derivatives and their applications A function is discontinuous at a point or has a discontinuity at a point if it is not continuous at the point lim x → a f ( x ) = L if for every ε >0, there exists a δ >0 such that if 0< | x − a | < δ , then | f ( x )− L | < ε An infinite discontinuity occurs at a point a if lim x → a − f ( x ) =±∞ or lim x → a + f ( x ) =±∞ A function has an infinite limit at a point a if it either increases or decreases without bound as it approaches a The instantaneous velocity of an object with a position function that is given by s ( t ) is the value that the average velocities on intervals of the form [ t , a ] and [ a , t ] approach as the values of t move closer to a , provided such a value exists the study of integrals and their applications Let f be continuous over a closed bounded interval ⎡ ⎣ a, b ⎤ ⎦ ; if z is any real number between f ( a ) and f ( b ), then there is a number c in ⎡ ⎣ a , b ⎤ ⎦ satisfying f ( c ) = z If all values of the function f ( x ) approach the real number L as the values of x ( ≠ a ) approach a , f ( x ) approaches L A jump discontinuity occurs at a point a if lim x → a − f ( x ) and lim x → a + f ( x ) both exist, but lim x → a − f ( x ) ≠ lim x → a + f ( x ) the process of letting x or t approach a in an expression; the limit of a function f ( x ) as x approaches a is the value
constant multiple law for limits
continuity at a point
continuity from the left
continuity from the right
continuity over an interval
difference law for limits
differential calculus discontinuity at a point
epsilon-delta definition of the limit
infinite discontinuity
infinite limit
instantaneous velocity
integral calculus Intermediate Value Theorem
intuitive definition of the limit
jump discontinuity
limit
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