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Math Help - Need Help WIth some questions on limits

  1. #1
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    Exclamation Need Help WIth some questions on limits

    1) lim x->INFINITY
    arctan(x^6 - x^8)
    the answer is -pi/2, but i dont know how to get it

    2) on pic (it will have a red number 2 on it)

    3) on second pic (it will have a red number 3 on it)

    for 2 and 3, both answers are on it, but i have no idea into how to get this
    does anyone know?????????
    Attached Thumbnails Attached Thumbnails Need Help WIth some questions on limits-math-1.jpg   Need Help WIth some questions on limits-math-2.jpg  
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  2. #2
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    Quote Originally Posted by Sneaky View Post
    1) lim x->INFINITY
    arctan(x^6 - x^8)
    the answer is -pi/2, but i dont know how to get it


    == Arctan is the inverse function of tan and tan x --> 00 if x --> Pi/2, and tan x --> -oo if x --> -Pi/2 (this is basic trigonometry, nothing fancy).
    From here what you get.


    2) on pic (it will have a red number 2 on it)


    == Just apply the def. of derivative to the function f(x) = 2^x at the point x = 4.


    3) on second pic (it will have a red number 3 on it)

    in case the limt exists, we know f ' (0) = lim [f(x) - f(0)]/x when x --> 0, so:

    lim [x sin(8/x) - 0]/x = lim sin(8/x) , and this doesn't exist since 8/x --> oo when x --> 0 from the right, and thus sin(8/x) swings between -1 and 1 without approaching any definite number (for example, choose x_n = 1/nPi so that 8/x_n = 8nPi, n an integer ==> sin(8/x_n) = 0 and clearly x_n --> 0 when n --> oo. Now choose y_n = 1/(n/16)Pi, with n an odd integer. again, y_n --> o when n-->oo, but this time sin(8/y_n) = (n/2)Pi, and since n is odd sin(8/y_n) is 1 or -1 ==> again, the limit doesn't exist.

    Tonio


    for 2 and 3, both answers are on it, but i have no idea into how to get this
    does anyone know?????????
    ...............................
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  3. #3
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    can you elaborate more on pic two, (the one with 2^x)
    when i derive 2^x i get:
    (ln2)( e^xln2)
    when i sub in 4 i get:
    16ln2

    for evaluating the top limit
    i am lost because there is a 2^x
    i can simplify the bottum to get
    (x-2)(x+2)
    which will make the equation 0/12 -> 0

    16ln2 does not equal 0
    so i still dont see why this one is the answer....
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  4. #4
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    Quote Originally Posted by Sneaky View Post
    can you elaborate more on pic two, (the one with 2^x)
    when i derive 2^x i get:
    (ln2)( e^xln2)
    when i sub in 4 i get:
    16ln2

    for evaluating the top limit
    i am lost because there is a 2^x
    i can simplify the bottum to get
    (x-2)(x+2)
    which will make the equation 0/12 -> 0

    16ln2 does not equal 0
    so i still dont see why this one is the answer....
    What's the definition of a function's derivative?
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  5. #5
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    derivitive is slope at a point x
    isnt it the (f(x+h)-f(x))h^-1
    when i use that i get:

    2^(x+h) - 2^x
    --------------
    h

    i dont know how to further simplify this...
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  6. #6
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    Quote Originally Posted by Sneaky View Post
    derivitive is slope at a point x
    isnt it the (f(x+h)-f(x))h^-1
    when i use that i get:

    2^(x+h) - 2^x
    --------------
    h

    i dont know how to further simplify this...
    You do not have to differentiate anything. You're simply asked to identify the function and the value of a. This is easily done by comparing the expression with the definition f'(a) = \lim_{x \rightarrow a} \frac{f(x) - f(a)}{x - a}. Surely you can make the comparison and identify what is f(x) and what is a ....
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