Denition approach of learning new topics

12 3 EXAMPLES OF WHAT IS A LIMIT?
We used above lim
θ→0
sin θ
θ
= 1 [1]
So we are proving this now!
In the neighbourhood of θ = 0. We see that
Area of ∆QRP ≤Area of sector QRP≤Area of ∆QRK
1
2 r2 sin θ ≤ 1 2 r2 θ ≤ 1 2 r2 tan θ
⇒ sin θ ≤ θ ≤ tan θ
Assuming we are seeing for θ > 0 without loss of generality, similar steps
and idea will follow for θ < 0
⇒ 1 ≤
θ
sin θ ≤ cos θ
⇒ cos θ ≤ sin θ
θ
≤ 1
sin θ
As θ → 0 ⇒ lim cos θ ≤ lim
θ→0 θ→0
θ
≤ 1
sin θ
⇒ lim
θ→0
θ
= 1 [Using Sandwich theorem]
3.2 Bucket is full or overowing?[5]
Consider this example of lling a bucket with the following rule, Each hour
the bucket is lled by half the amount left empty. So when the process is
started at the beginning the bucket is empty. First hour half the bucket is
half lled ( 1 2 ) , next hour we have half bucket left and hence we ll ( 1 2 + 1 4 ).
Third hour we have lled half of the left lled, So we have ( 1 2 + 1 4 + 1 8 ). And
keep doing this process we get ( 1 2 + 1 4 + 1 8 + · · · = 1/2
1−1/2 = 1).
So this is another example of limit concept. Here the end state of this process
is that the bucket is always lled by half the amount left and that would never
allow the bucket to get full, but if the process is continued till innity the end
state would be lling the bucket.