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Thread: 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 $\displaystyle a$. This is easily done by comparing the expression with the definition $\displaystyle f'(a) = \lim_{x \rightarrow a} \frac{f(x) - f(a)}{x - a}$. Surely you can make the comparison and identify what is $\displaystyle f(x) $ and what is $\displaystyle a$ ....
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