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Homework Help: Finding limits in differentiation from first principles

  1. Apr 7, 2012 #1
    1. The problem statement, all variables and given/known data

    Differentiate sin(ax), cos(ax) and tan(ax) from first principles.

    2. Relevant equations

    3. The attempt at a solution

    I have used first principles to differentiate the three expressions and have been successful until I encountered limits of some expressions in the process.

    I need to find the limit as Δx tends to 0 of the following expressions.

    1. sin(aΔx)/Δx
    2. [cos(Δx) - 1]/Δx
    3. tan(Δx)/Δx

    I know some spooky proofs which use the fact that for small Δx, sin(aΔx) ≈ Δx, cos(aΔx) ≈ 1- (aΔx)2/2 and tan(aΔx) ≈ Δx. They do give the right answers and I have been told these methods would give me full marks in the exam (me being a physics student and all that crap!), but I would appreciate it if you give a full rigorous proof of the three limits. (Armed with those, the original problem is just a piece of cake.)
  2. jcsd
  3. Apr 7, 2012 #2


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    hi failexam :smile:

    (rsinθ)/rθ = arc-length/chord-length :wink:
  4. Apr 7, 2012 #3

    Ray Vickson

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    Once you have [tex] \lim_{t \rightarrow 0} \frac{\sin(t)}{t} = 1,[/tex]
    [tex] \frac{1 - \cos(t)}{t^2} \rightarrow \frac{1}{2} \text{ as } t \rightarrow 0[/tex] follows easily from [tex] \frac{\sin(t)^2}{t^2} = \frac{1 - \cos(t)^2}{t^2}
    = \frac{1-\cos(t)}{t^2} (1+\cos(t)),[/tex] and the limit of [itex] \tan(t)/t[/itex] also follows. So, you need a good proof of [itex] \sin(t)/t \rightarrow 1.[/itex] You can find one in the Khan Academy video .

    Last edited by a moderator: Sep 25, 2014
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