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AP Calculus BC Units 2-6 Notes
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Chapters 2-6 cover the following: - Limits and Derivatives - Differentiation Rules - Applications of Differentiation - Integrals - Applications of Integration
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Unit 2 | Limits and DerivativesSection 2.2 | The Limit of a Function
Intuitive Definition of a Limit
Suppose f(x)is defined when xis near thew number a. (This means that f is defined
on some open interval that contains a, except possibly at aitself.) Then we write
limx→a f(x) = L and say “the limit of f(x), as xapproaches a, equals L” if we can
make the values of f(x)arbitrarily close to L(as close to Las we like) by restricting x
to be sufficiently close to a(on either side of a) but not equal to a.
Definition of One-Sided Limits
We write limx→a− f(x) = Land say the left-hand limit of f (x)as xapproaches a[or
the limit of f (x)as xapproacjes afrom the left] is equal to Lif we can mkae the
values of f(x)arbitrarily close to Lby taking xto be sufficiently close to awith xless
than a.
Unit 2 | Limits and Derivatives 1
, lim f(x) = L
x→a
if and only if
lim f(x) = L
x→a−
and
lim f(x) = L
x→a+
Intuitive Definition of an Infinite Limit and Infinite Limits
Let f be a function defined on both sides of a, except possibly at aitself. Then
limx→a f(x) = ∞ means that the values of f(x)can be made arbitrarily large (as
large as we please) by taking xsufficiently close to a, but not equal to a.
Let f be a function defined on both sides of a, except possibyl at aitself. Then
limx→a f(x) = −∞ means that the values of f(x)can be made arbitrarily large by
taking xsufficiently close to a, but not equal to a.
The vertical line x = ais called a vertical asymptote of the curve y = f(x)if at least
one of the following statements is true:
lim f(x) = ∞
lim f(x) = ∞
lim f(x) = ∞
x→a x→a− x→a+
lim f(x) = −∞
lim f(x) = −∞
lim f(x) = −∞
x→a x→a− x→a−
Another example of a function whose graph has a vertical asymptote is the natural
logarithmic function y = ln x.
lim ln x = −∞
x→0+
Section 2.3 | Calculating Limits Using the Limit
Laws
Limit Laws
Unit 2 | Limits and Derivatives 2
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