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lecture10

# lecture10 - m then 1 If n< m 2 If n = m 3 If n> m ex...

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Lecture 10: Limits at Inﬂnity (Section 2.6) Consider the graph of f ( x ) = x 2 x 2 + 1 : What happens to f ( x ) as x gets larger in a positive or negative direction? lim x !1 f ( x ) = lim x !¡1 f ( x ) =

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Def. Let f be deﬂned on some interval ( a; 1 ). Then lim x !1 f ( x ) = L means that the values of f ( x ) can be made arbitrarily close to L by taking x su–- ciently large. f ( x ) as x Def. The line y = L is called a of the curve if either lim x !1 f ( x ) = L or lim x !¡1 f ( x ) = L
How many horizontal asymptotes can a graph have? Consider the following functions: ex. f ( x ) = x 2 ex. f ( x ) = e x 6 - ? ± 6 - ? ± ex. f ( x ) = tan ¡ 1 ( x ) 6 - ? ± lim x !1 tan ¡ 1 ( x ) = lim x !¡1 tan ¡ 1 ( x ) =

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Theorem: If r > 0 is a rational number, then lim x !1 1 x r = lim x !¡1 1 x r = ex. lim x !1 x 2 + 2 x x 3 + 3 x =
ex. lim x !¡1 x 4 ¡ 1 1 ¡ x 3 = ex. lim x !¡1 3 x (2 x 2 + 1) 6 ¡ x 2 ¡ 2 x 3 =

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Shortcut for ﬂnding limits at inﬂnity for rational functions If f ( x ) = p ( x ) q ( x ) where p ( x ) is of degree n and q ( x ) is of degree

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Unformatted text preview: m , then 1) If n < m 2) If n = m 3) If n > m ex. Find the domain and horizontal asymptotes of f ( x ) = 3 x p 4 x 2 Â¡ 8 Evaluate the inï¬‚nite limits: ex. lim x !1 ( x 2 Â¡ x 3 ) ex. lim x !1 sin( x ) x Consider the following limits involving exponentials: 6-? Â± ex. lim x !1 e x = ex. lim x !1 e Â¡ x = ex. lim x !Â¡1 e 1 x = ex. Find the vertical and horizontal asymptotes of y = 2 2 + 4 e Â¡ x . ex. Sketch by ï¬‚nding intercepts and limits: y = x ( x + 2) 2 (1 Â¡ x ) 3 6-? Â± Additional Examples ex. lim x !1 s 5 Â¡ x 2 2 Â¡ x 2 ex. lim x !1 3 e x 2 e x Â¡ 3 ex. lim x !1 p x 2 Â¡ 2 x Â¡ x...
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