# Homework Help: Finding limits in differentiation from first principles

1. Apr 7, 2012

### spaghetti3451

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. Apr 7, 2012

### tiny-tim

hi failexam

(rsinθ)/rθ = arc-length/chord-length

3. Apr 7, 2012

### Ray Vickson

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

RGV

Last edited by a moderator: Sep 25, 2014