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Math Help - trig limits

  1. #1
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    trig limits

    limit t --> 0 of : (tan(pie/4 + t) - tan(pie/4)) / t

    I convert to sins and cosines, try to evaluate and get 1 sometimes, 0 other times. I try to expand out the sin and cos of (pie/4+t) but still get the same answers. I try to multiply both sides by sin(T)/sin(T) and t/t but its messed up because sin(T)= 0 so I usually end up with 1 on the bottom making either the whole expression void or it ends up as sqrt(2)/2 / sqrt(2)/2 which is 1.

    So can anyone help? im probably making some big mistakes here, but i mean I think ive been at it for a few hours already......

    also the test says the answer = 2
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  2. #2
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    e^(i*pi)'s Avatar
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    Have you tried L'hopital's rule?
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  3. #3
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    Quote Originally Posted by frankinaround View Post
    limit t --> 0 of : (tan(pie/4 + t) - tan(pie/4)) / t
    By the way, you spell it "pi."

    -Dan
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  4. #4
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    Hello, frankinaround!


    . . . . . . tan(π/4 + t) - tan(π/4)
    . . lim .---------------------------
    . t →0 . . . . . . . t

    . . . . . . . . . . . . . .tan(π/4) + tan(t)
    The numerator is: .--------------------- .- .tan(π/4)
    . . . . . . . . . . . . . .1 - tan(π/4)tan(t)

    . . . . . . . . . . . . . . 1 + tan(t)
    . . . . . . . . . . . = . ------------- .- .1
    . . . . . . . . . . . . . . 1 - tan(t)

    . . . . . . . . . . . . . . 2 tan(t)
    . . . . . . . . . . . = . ------------
    . . . . . . . . . . . . . . 1 - tan(t)

    . . . . . . . . . . . . . . . sin(t)
    . . . . . . . . . . . . . . 2 -------
    . . . . . . . . . . . . . . . cos(t)
    . . . . . . . . . . . = .--------------
    . . . . . . . . . . . . . . . .sin(t)
    . . . . . . . . . . . . . 1 - -------
    . . . . . . . . . . . . . . . .cos(t)


    Multiply numerator and denominator by cos(t):

    . . . . .2 sin(t)
    . . ------------------
    . . .cos(t) - sin(t)

    . . . . . . . . . . . . .2 sin(t). . . . . . . . . . . . 2. . . . . . sin(t)
    Divide by t: . -------------------- . = . ---------------- . --------
    . . . . . . . . . . t[cos(t) - sin(t)]. . . . cos(t) - sin(t). . . .t


    . . . . . . . . . . . . . . . 2 . . . . . . sin(t) . . . . . 2
    Therefore: .lim .---------------- . ------- . = . ------ . 1 . = . 2
    . . . . . . . . t→0.cos(t) - sin(t) . . . t . . . . . 1 - 0


    [When will LaTeX be restored?]

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