limit rules - rmlowman/math165/lecturenotes/l5-w2l2r · limit rules example lim x!3 x2 9 x 3 =?...

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  • Limit RulesUseful rules for finding limits:

    In the following rules assume k = constant

    limxc

    k = k

    limxc

    kf(x) = k limxc

    f(x)

    limxc

    f(x) g(x) = limxc

    f(x) limxc

    g(x)

    limxc

    f(x) g(x) = limxc

    f(x) limxc

    g(x)

    limxc

    f(x)

    g(x)=

    limxc f(x)

    limxc g(x)( 6=0)

    limxc

    [f(x)n] =[limxc

    f(x)]n

    limxc

    P(x) = P(c) , P(x) is continuous around c, just plug-in

  • Limit RulesUseful rules for finding limits:In the following rules assume k = constant

    limxc

    k = k

    limxc

    kf(x) = k limxc

    f(x)

    limxc

    f(x) g(x) = limxc

    f(x) limxc

    g(x)

    limxc

    f(x) g(x) = limxc

    f(x) limxc

    g(x)

    limxc

    f(x)

    g(x)=

    limxc f(x)

    limxc g(x)( 6=0)

    limxc

    [f(x)n] =[limxc

    f(x)]n

    limxc

    P(x) = P(c) , P(x) is continuous around c, just plug-in

  • Limit RulesUseful rules for finding limits:In the following rules assume k = constant

    limxc

    k = k

    limxc

    kf(x) = k limxc

    f(x)

    limxc

    f(x) g(x) = limxc

    f(x) limxc

    g(x)

    limxc

    f(x) g(x) = limxc

    f(x) limxc

    g(x)

    limxc

    f(x)

    g(x)=

    limxc f(x)

    limxc g(x)( 6=0)

    limxc

    [f(x)n] =[limxc

    f(x)]n

    limxc

    P(x) = P(c) , P(x) is continuous around c, just plug-in

  • Limit RulesUseful rules for finding limits:In the following rules assume k = constant

    limxc

    k = k

    limxc

    kf(x) = k limxc

    f(x)

    limxc

    f(x) g(x) = limxc

    f(x) limxc

    g(x)

    limxc

    f(x) g(x) = limxc

    f(x) limxc

    g(x)

    limxc

    f(x)

    g(x)=

    limxc f(x)

    limxc g(x)( 6=0)

    limxc

    [f(x)n] =[limxc

    f(x)]n

    limxc

    P(x) = P(c) , P(x) is continuous around c, just plug-in

  • Limit RulesUseful rules for finding limits:In the following rules assume k = constant

    limxc

    k = k

    limxc

    kf(x) = k limxc

    f(x)

    limxc

    f(x) g(x) = limxc

    f(x) limxc

    g(x)

    limxc

    f(x) g(x) = limxc

    f(x) limxc

    g(x)

    limxc

    f(x)

    g(x)=

    limxc f(x)

    limxc g(x)( 6=0)

    limxc

    [f(x)n] =[limxc

    f(x)]n

    limxc

    P(x) = P(c) , P(x) is continuous around c, just plug-in

  • Limit RulesUseful rules for finding limits:In the following rules assume k = constant

    limxc

    k = k

    limxc

    kf(x) = k limxc

    f(x)

    limxc

    f(x) g(x) = limxc

    f(x) limxc

    g(x)

    limxc

    f(x) g(x) = limxc

    f(x) limxc

    g(x)

    limxc

    f(x)

    g(x)=

    limxc f(x)

    limxc g(x)( 6=0)

    limxc

    [f(x)n] =[limxc

    f(x)]n

    limxc

    P(x) = P(c) , P(x) is continuous around c, just plug-in

  • Limit RulesUseful rules for finding limits:In the following rules assume k = constant

    limxc

    k = k

    limxc

    kf(x) = k limxc

    f(x)

    limxc

    f(x) g(x) = limxc

    f(x) limxc

    g(x)

    limxc

    f(x) g(x) = limxc

    f(x) limxc

    g(x)

    limxc

    f(x)

    g(x)=

    limxc f(x)

    limxc g(x)( 6=0)

    limxc

    [f(x)n] =[limxc

    f(x)]n

    limxc

    P(x) = P(c) , P(x) is continuous around c, just plug-in

  • Limit RulesUseful rules for finding limits:In the following rules assume k = constant

    limxc

    k = k

    limxc

    kf(x) = k limxc

    f(x)

    limxc

    f(x) g(x) = limxc

    f(x) limxc

    g(x)

    limxc

    f(x) g(x) = limxc

    f(x) limxc

    g(x)

    limxc

    f(x)

    g(x)=

    limxc f(x)

    limxc g(x)( 6=0)

    limxc

    [f(x)n] =[limxc

    f(x)]n

    limxc

    P(x) = P(c) , P(x) is continuous around c, just plug-in

  • Limit RulesUseful rules for finding limits:In the following rules assume k = constant

    limxc

    k = k

    limxc

    kf(x) = k limxc

    f(x)

    limxc

    f(x) g(x) = limxc

    f(x) limxc

    g(x)

    limxc

    f(x) g(x) = limxc

    f(x) limxc

    g(x)

    limxc

    f(x)

    g(x)=

    limxc f(x)

    limxc g(x)( 6=0)

    limxc

    [f(x)n] =[limxc

    f(x)]n

    limxc

    P(x) = P(c) , P(x) is continuous around c, just plug-in

  • Limit Rulesexample

    limx3

    x2 9x 3

    =?

    first try limit of ratio = ratio of limits rule,

    limx3

    x2 9x 3

    =limx3 x2 9limx3 x 3

    =0

    0

    00 is called an indeterminant form. When you reach an indeterminant formyou need to try someting else.

    limx3

    x2 9x 3

    = limx3

    (x 3)(x + 3)(x 3)

    = limx3

    (x + 3) = 3 + 3 = 6

    Indeterminant does not mean the limit does not exist.It just means that the method you tried did not tell you anythingand you need to try another method.

  • Limit Rulesexample

    limx3

    x2 9x 3

    =?

    first try limit of ratio = ratio of limits rule,

    limx3

    x2 9x 3

    =limx3 x2 9limx3 x 3

    =0

    0

    00 is called an indeterminant form. When you reach an indeterminant formyou need to try someting else.

    limx3

    x2 9x 3

    = limx3

    (x 3)(x + 3)(x 3)

    = limx3

    (x + 3) = 3 + 3 = 6

    Indeterminant does not mean the limit does not exist.It just means that the method you tried did not tell you anythingand you need to try another method.

  • Limit Rulesexample

    limx3

    x2 9x 3

    =?

    first try limit of ratio = ratio of limits rule,

    limx3

    x2 9x 3

    =

    limx3 x2 9limx3 x 3

    =0

    0

    00 is called an indeterminant form. When you reach an indeterminant formyou need to try someting else.

    limx3

    x2 9x 3

    = limx3

    (x 3)(x + 3)(x 3)

    = limx3

    (x + 3) = 3 + 3 = 6

    Indeterminant does not mean the limit does not exist.It just means that the method you tried did not tell you anythingand you need to try another method.

  • Limit Rulesexample

    limx3

    x2 9x 3

    =?

    first try limit of ratio = ratio of limits rule,

    limx3

    x2 9x 3

    =limx3 x2 9limx3 x 3

    =

    0

    0

    00 is called an indeterminant form. When you reach an indeterminant formyou need to try someting else.

    limx3

    x2 9x 3

    = limx3

    (x 3)(x + 3)(x 3)

    = limx3

    (x + 3) = 3 + 3 = 6

    Indeterminant does not mean the limit does not exist.It just means that the method you tried did not tell you anythingand you need to try another method.

  • Limit Rulesexample

    limx3

    x2 9x 3

    =?

    first try limit of ratio = ratio of limits rule,

    limx3

    x2 9x 3

    =limx3 x2 9limx3 x 3

    =0

    0

    00 is called an indeterminant form. When you reach an indeterminant formyou need to try someting else.

    limx3

    x2 9x 3

    = limx3

    (x 3)(x + 3)(x 3)

    = limx3

    (x + 3) = 3 + 3 = 6

    Indeterminant does not mean the limit does not exist.It just means that the method you tried did not tell you anythingand you need to try another method.

  • Limit Rulesexample

    limx3

    x2 9x 3

    =?

    first try limit of ratio = ratio of limits rule,

    limx3

    x2 9x 3

    =limx3 x2 9limx3 x 3

    =0

    0

    00 is called an indeterminant form.

    When you reach an indeterminant formyou need to try someting else.

    limx3

    x2 9x 3

    = limx3

    (x 3)(x + 3)(x 3)

    = limx3

    (x + 3) = 3 + 3 = 6

    Indeterminant does not mean the limit does not exist.It just means that the method you tried did not tell you anythingand you need to try another method.

  • Limit Rulesexample

    limx3

    x2 9x 3

    =?

    first try limit of ratio = ratio of limits rule,

    limx3

    x2 9x 3

    =limx3 x2 9limx3 x 3

    =0

    0

    00 is called an indeterminant form. When you reach an indeterminant formyou need to try someting else.

    limx3

    x2 9x 3

    = limx3

    (x 3)(x + 3)(x 3)

    = limx3

    (x + 3) = 3 + 3 = 6

    Indeterminant does not mean the limit does not exist.It just means that the method you tried did not tell you anythingand you need to try another method.

  • Limit Rulesexample

    limx3

    x2 9x 3

    =?

    first try limit of ratio = ratio of limits rule,

    limx3

    x2 9x 3

    =limx3 x2 9limx3 x 3

    =0

    0

    00 is called an indeterminant form. When you reach an indeterminant formyou need to try someting else.

    limx3

    x2 9x 3

    =

    limx3

    (x 3)(x + 3)(x 3)

    = limx3

    (x + 3) = 3 + 3 = 6

    Indeterminant does not mean the limit does not exist.It just means that the method you tried did not tell you anythingand you need to try another method.

  • Limit Rulesexample

    limx3

    x2 9x 3

    =?

    first try limit of ratio = ratio of limits rule,

    limx3

    x2 9x 3

    =limx3 x2 9limx3 x 3

    =0

    0

    00 is called an indeterminant form. When you reach an indeterminant formyou need to try someting else.

    limx3

    x2 9x 3

    = limx3

    (x 3)(x + 3)(x 3)

    =

    limx3

    (x + 3) = 3 + 3 = 6

    Indeterminant does not mean the limit does not exist.It just means that the method you tried did not tell you anythingand you need to try another method.

  • Limit Rulesexample

    limx3

    x2 9x 3

    =?

    first try limit of ratio = ratio of limits rule,

    limx3

    x2 9x 3

    =limx3 x2 9limx3 x 3

    =0

    0

    00 is called an indeterminant form. When you reach an indeterminant formyou need to try someting else.

    limx3

    x2 9x 3

    = limx3

    (x 3)(x +

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