# Finding limits in differentiation from first principles

• spaghetti3451
Proof of the Limit of Sin and Cosine from First Principles:https://www.youtube.com/watch?v=NbvG3P3_B58In summary, using first principles, the author was able to find the limits of sin(ax) and cos(ax) as Δx tends to 0. They were able to do this using the fact that sin(aΔx)/Δx and cos(aΔx)/Δx are equal, and that tan(aΔx)/Δx is equal to Δx. They also found the limit of sin(ax) as Δx tends to 0 using the spooky proof mentioned in the article. Finally,

## Homework Statement

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

## 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.)

hi failexam

(rsinθ)/rθ = arc-length/chord-length

failexam said:

## Homework Statement

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

## 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.)

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

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