Chapter 2 | Limits
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limit laws multivariable calculus one-sided limit power law for limits that f ( x ) approaches as x approaches a the individual properties of limits; for each of the individual laws, let f ( x ) and g ( x ) be defined for all x ≠ a over some open interval containing a ; assume that L and M are real numbers so that lim x → a f ( x ) = L and lim x → a g ( x ) = M ; let c be a constant the study of the calculus of functions of two or more variables A one-sided limit of a function is a limit taken from either the left or the right the limit law lim x → a ⎛ ⎝ f ( x ) ⎞ ⎠ n = ⎛ ⎝ lim x → a f ( x ) ⎞ ⎠ n = L n for every positive integer n
the limit law lim x → a ⎛
⎝ f ( x ) · g ( x ) ⎞
f ( x ) · lim x → a
g ( x ) = L · M
⎠ = lim x → a
product law for limits
f ( x )
lim x → a lim x → a
quotient law for limits
f ( x ) g ( x ) =
L M for
the limit law lim x → a
M ≠0
g ( x ) =
A removable discontinuity occurs at a point a if f ( x ) is discontinuous at a , but lim x → a f ( x )
removable discontinuity
exists
= L n
f ( x ) n = lim x → a n
root law for limits
f ( x )
the limit law lim x → a
for all L if n is odd and for L ≥0 if n is even
A secant line to a function f ( x ) at a is a line through the point ⎛
⎝ a , f ( a ) ⎞
secant
⎠ and another point on the function; the
f ( x )− f ( a ) x − a states that if f ( x ) ≤ g ( x ) ≤ h ( x ) for all x ≠ a over an open interval containing a and
slope of the secant line is given by m sec =
squeeze theorem
f ( x ) = L = lim x → a
h ( x ) where L is a real number, then lim x → a
g ( x ) = L
lim x → a
The limit law lim x → a ⎛
⎝ f ( x )+ g ( x ) ⎞
f ( x )+ lim x → a
g ( x ) = L + M
⎠ = lim x → a
sum law for limits
tangent ⎝ a , f ( a ) ⎞ ⎠ approach as they are taken through points on the function with x -values that approach a ; the slope of the tangent line to a graph at a measures the rate of change of the function at a If a and b are any real numbers, then | a + b | ≤ | a | + | b | A function has a vertical asymptote at x = a if the limit as x approaches a from the right or left is infinite KEY EQUATIONS • Slope of a Secant Line m sec = f ( x )− f ( a ) x − a • Average Velocity over Interval [ a , t ] v ave = s ( t )− s ( a ) t − a • Intuitive Definition of the Limit lim x → a f ( x ) = L • Two Important Limits lim x → a x = a lim x → a c = c • One-Sided Limits triangle inequality vertical asymptote A tangent line to the graph of a function at a point ⎛ ⎝ a , f ( a ) ⎞ ⎠ is the line that secant lines through ⎛
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