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Chapter 2 | Limits
f ( x ) = L
f ( x ) = L
lim x → a −
lim x → a +
• Infinite Limits from the Left lim x → a − f ( x ) =+∞ lim x → a − • Infinite Limits from the Right lim x → a + f ( x ) =+∞ lim x → a + • Two-Sided Infinite Limits lim x → a f ( x ) =+∞: lim x → a −
f ( x ) =−∞
f ( x ) =−∞
f ( x ) =+∞ and lim x → a + f ( x ) =−∞ and lim x → a +
f ( x ) =+∞
f ( x ) =−∞: lim x → a −
f ( x ) =−∞
lim x → a
• Basic Limit Results lim x → a x = a lim x → a c = c • Important Limits lim θ →0 sin θ =0 lim θ →0 cos θ =1 lim θ →0 sin θ θ =1 lim θ →0 1−cos θ θ =0
KEY CONCEPTS 2.1 A Preview of Calculus
• Differential calculus arose from trying to solve the problem of determining the slope of a line tangent to a curve at a point. The slope of the tangent line indicates the rate of change of the function, also called the derivative . Calculating a derivative requires finding a limit. • Integral calculus arose from trying to solve the problem of finding the area of a region between the graph of a function and the x -axis. We can approximate the area by dividing it into thin rectangles and summing the areas of these rectangles. This summation leads to the value of a function called the integral . The integral is also calculated by finding a limit and, in fact, is related to the derivative of a function. • Multivariable calculus enables us to solve problems in three-dimensional space, including determining motion in space and finding volumes of solids. 2.2 The Limit of a Function • A table of values or graph may be used to estimate a limit. • If the limit of a function at a point does not exist, it is still possible that the limits from the left and right at that point may exist. • If the limits of a function from the left and right exist and are equal, then the limit of the function is that common value. • We may use limits to describe infinite behavior of a function at a point. 2.3 The Limit Laws • The limit laws allow us to evaluate limits of functions without having to go through step-by-step processes each time. • For polynomials and rational functions, lim x → a f ( x ) = f ( a ).
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