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Math Help - cal 1 problems limits

  1. #1
    Junior Member Cyberman86's Avatar
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    cal 1 problems limits

    I can't figure out this one, its driving me crazy. My teacher made this problems up and he call them Interesting problems and sometimes even the tutors struggle with his problems.

    Lim 5 + 2x^2 + x^3 / over 3 - x^2 + 2x^3
    x->infinity


    It has the sign of Infinity but i can't draw it and -> is an arrow as x approaches...

    ---------------------------------------------------------------------------

    Evaluate the limit and illustrate each step that involves the use of a limit property.

    Lim t^3 - 3t^2 - t + 3 / over t^2 - t - 6
    t->3

    as t approaches 3

    so far i have:

    -> 27 - 27 - 3 +3 / over 9 - 3 - 6

    -> 0 / over 0

    = 0

    I substitute it 3 for t but i don't know if this is the right way for this one, Also i need to illustrate each step that involves the use of a limit property, but how? Do i have to label each step like by hypothesis, by substitution, by arithmetic etc..?
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  2. #2
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    divide every term in the quotient by the highest power of x (that would be x^3).

    now, what happens to the value of the expression as x gets large?
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  3. #3
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    Quote Originally Posted by Cyberman86 View Post
    I can't figure out this one, its driving me crazy. My teacher made this problems up and he call them Interesting problems and sometimes even the tutors struggle with his problems.

    Lim 5 + 2x^2 + x^3 / over 3 - x^2 + 2x^3
    x->infinity

    It has the sign of Infinity but i can't draw it and -> is an arrow as x approaches...

    [snip]
    There are numerous approaches. I'll use the simplest:

     \lim_{x \rightarrow + \infty} \frac{x^3 + 2x^2 + 5}{2x^3 - x^2 + 3}

    Divide numerator and denominator by x^3:

     = \lim_{x \rightarrow + \infty} \frac{1 + \frac{2}{x} + \frac{5}{x^3}}{2 - \frac{1}{x} + \frac{3}{x^3}} = \frac{1 + 0 + 0}{2 - 0 + 0} = \frac{1}{2}.
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  4. #4
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    Quote Originally Posted by Cyberman86 View Post
    [snip]
    Evaluate the limit and illustrate each step that involves the use of a limit property.

    Lim t^3 - 3t^2 - t + 3 / over t^2 - t - 6
    t->3
    as t approaches 3

    so far i have:

    -> 27 - 27 - 3 +3 / over 9 - 3 - 6

    -> 0 / over 0

    = 0

    I substitute it 3 for t but i don't know if this is the right way for this one, Also i need to illustrate each step that involves the use of a limit property, but how? Do i have to label each step like by hypothesis, by substitution, by arithmetic etc..?
     \lim_{t \rightarrow 3} \frac{t^3 - 3t^2 - t + 3}{t^2 - t - 6} = \lim_{t \rightarrow 3} \frac{t^2 (t - 3) - (t - 3)}{(t-3)(t+2)}.

    Finish factorising the numerator, cancel the factor common to numerator and denominator and take the limit.
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  5. #5
    Junior Member Cyberman86's Avatar
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    did i finish it correctly:

    9 - (3 - 3) / over (3 + 2)

    = 9/5
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  6. #6
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    Quote Originally Posted by Cyberman86 View Post
    did i finish it correctly:

    9 - (3 - 3) / over (3 + 2)

    = 9/5
    No.

    Show all your working in factorising t^2 (t - 3) - (t - 3). Then either someone can show you the mistake(s) you've made or you might actually realise the mistake yourself.
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  7. #7
    Junior Member Cyberman86's Avatar
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    Unhappy

    here i go again:

    -> t^2 (t-3) - (t-3) / (t-3) (t+2)

    -> numerator and denominator (t-3) cancel

    -> t^2 - (t-3) / (t+2)

    -> then i substitute t for 3?

    -> 3^2 - (3-3) / (3+2)

    -> 9 - (3-3) / (3+2)

    -> 9 / 5


    i showed all i did but still i don't know if that's what you mean
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  8. #8
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    Quote Originally Posted by Cyberman86 View Post
    here i go again:

    -> t^2 (t-3) - (t-3) / (t-3) (t+2)

    -> numerator and denominator (t-3) cancel

    -> t^2 - (t-3) / (t+2)

    -> then i substitute t for 3?

    -> 3^2 - (3-3) / (3+2)

    -> 9 - (3-3) / (3+2)

    -> 9 / 5


    i showed all i did but still i don't know if that's what you mean
    Learn to factorise properly and you'll probably have greater success with questions that depend on that skill.

    t^2 (t - 3) - (t - 3)

    Take out the common factor:

    = (t - 3) (t^2 - 1).
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