Proof of limit of sine theta as theta approaches zero

This is a proof of `lim_(θ -> 0) sin(θ) = 0`

Consider a right angled triangle ABC. Let angle C = θ

In the above triangle, sine theta is defined as

`sin(θ) = "Opposite" / "Hypotenuse" = "AB"/"AC"`

Thus the limit of sin(θ) as θ approaches zero is

`lim_(θ -> 0) sin(θ) = lim_(θ -> 0) "AB"/"AC"`

As theta becomes smaller so does the opposite side AB. Thus as theta approaches zero, AB approaches zero as well, and when theta equals zero, AB equals zero as well. Thus the limit of sine theta as theta approaches zero is,

`lim_(θ -> 0) sin(θ) = lim_(θ -> 0) "AB"/"AC" = 0/"AC" = 0`

Since zero divided by anything is zero so 0/AC is equal to zero, and hence the limit of sine theta as theta approaches zero is equal to zero, that is,

`lim_(θ -> 0) sin(θ) = 0`

QED