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Math Help - Find the Inidicated Limit

  1. #1
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    Find the Inidicated Limit

    Find the indicated Limit, if the limit does not exist, so state that, or use the symbol (infinity) or (-infinity) where appropriate.

    There one thing I don't understand from reading my textbook and I'll explain after I post the questions.

    Question 1: f(x) = {2 if x <or= 2
    {1 if x > 2

    a) Find lim x-->2+(from the right) = 1
    b) Find lim x--> 2-(from left) = 2
    c) Find lim x-->2 (the limit does not exist since Lim 2+ and lim 2- are differnet)
    d) Find lim x--> infinity = 1
    e) Find lim x--> - infinity = 2

    Ok so the above question I understand, now my question as pertaining to this post is the following (is related to Question 1):

    Question 2: g(x) = { x if x<0
    { -x if x>0

    a) lim x--> 0+ g(x) (from right)
    b) lim x--> 0- g(x) (from left)
    c) lim x-- 0 g(x)
    d) lim x--> infinity g(x)
    e) lim x--> - infinity g(x)

    now since I answered question one correctly I assumed you do the same for question 2 so my answers looked like this:

    a) = -x (since its from the right and greater than 0)
    b) = x
    c) the limit does not exist because 0- and 0+ are different
    d) = -x
    e) = x

    in my book however it says:

    a) 0
    b) 0
    c) 0
    d) -infinity
    e) -infinity

    I don't understand these answers since the equations at the top say that if x is greater or less than g(x) would be either x or -x .. why then would they put in values of 0? and -infinity and +infinity?
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by alisheraz19 View Post
    Find the indicated Limit, if the limit does not exist, so state that, or use the symbol (infinity) or (-infinity) where appropriate.

    There one thing I don't understand from reading my textbook and I'll explain after I post the questions.

    Question 1: f(x) = {2 if x <or= 2
    {1 if x > 2

    a) Find lim x-->2+(from the right) = 1
    b) Find lim x--> 2-(from left) = 2
    c) Find lim x-->2 (the limit does not exist since Lim 2+ and lim 2- are differnet)
    d) Find lim x--> infinity = 1
    e) Find lim x--> - infinity = 2

    Ok so the above question I understand, now my question as pertaining to this post is the following (is related to Question 1):

    Question 2: g(x) = { x if x<0
    { -x if x>0

    a) lim x--> 0+ g(x) (from right)
    b) lim x--> 0- g(x) (from left)
    c) lim x-- 0 g(x)
    d) lim x--> infinity g(x)
    e) lim x--> - infinity g(x)

    now since I answered question one correctly I assumed you do the same for question 2 so my answers looked like this:

    a) = -x (since its from the right and greater than 0)
    b) = x
    c) the limit does not exist because 0- and 0+ are different
    d) = -x
    e) = x

    in my book however it says:

    a) 0
    b) 0
    c) 0
    d) -infinity
    e) -infinity

    I don't understand these answers since the equations at the top say that if x is greater or less than g(x) would be either x or -x .. why then would they put in values of 0? and -infinity and +infinity?
    lets just consider (a) for the moment. question: what is \lim_{x \to 0^+} -x? Is it -x? you never actually took the limit! you just pointed out what function you would use. you got lucky with the first question because any limit of a constant function happens to be the function itself, the constant. that is not the case with non-constant functions.
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  3. #3
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    ok but why 0? if the question asks 0+ wouldn't the number have to be a close approximation to 0? such as -0.00001? or is it 0 because that's close enough to 0?
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  4. #4
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by alisheraz19 View Post
    ok but why 0? if the question asks 0+ wouldn't the number have to be a close approximation to 0? such as -0.00001? or is it 0 because that's close enough to 0?
    if a function is continuous where you are taking the limit, just plug in the value. You will get arbitrarily close to zero as much as you want when coming from the right. We will eventually be closer to zero than we are to -0.00001, so -0.00001 cannot be the limit, and this will be true for any nonzero number you choose. The limit is zero.

    \lim_{x \to 0^+} -x = -0 = 0
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