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# lecture10 - Lecture 10 Limits at Inﬂnity(Chapter 2...

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Lecture 10: Limits at Inﬂnity (Chapter 2, Section 10) Consider the graph of f ( x ) = x 2 x 2 + 1 : What happens to f ( x ) as x increases in absolute value? In other words, what is lim x !1 f ( x ) = lim x !¡1 f ( x ) =

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Def. Let function f : R ! R be deﬂned on some interval ( a; 1 ). We say that f ( x ) has limit L as x approaches 1 if the values of f ( x ) can be made arbitrarily close to L by taking x su–ciently large. We write lim x !1 f ( x ) = L . More formally, we say that lim x !1 f ( x ) = L if for any small positive number ² , we can ﬂnd a positive real number N so that j f ( x ) ¡ L j < ² for all x > N . We have similar deﬂnitions as x ! ¡1 . Def. Let f ( x ) be deﬂned on some interval ( ¡1 ;b ). We say lim x !¡1 f ( x ) = L if the values of f ( x ) can be made arbitrarily close to L by taking x to be negative but su–ciently large in absolute value. That is, for any small positive ² we can ﬂnd a negative real number N so that j f ( x ) ¡ L j < ² for all x < N .
Def. The line y = L is called a of the graph of f ( x ) if either lim x !1 f ( x ) = L or lim x !¡1 f ( x ) = L How many horizontal asymptotes can a graph have?

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lecture10 - Lecture 10 Limits at Inﬂnity(Chapter 2...

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