Limit and Continuity, Finding Limits of Some of the Important Functions

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Finding Limits of Some of the Important Functions

(3) Prove that

Proof:

Draw a circle of radius unit and with centre at the origin . Let ) be a point on the circle. Let be any other point on the circle. Draw

A circle of radius 1

A Circle of Radius 1

A circle of radius 1

Let radians, where

Draw a tangent to the circle at meeting produced at . Then .

Area of area of sector area of .

Or

i.e. [Dividing throughout by ]

Or

Or

i.e.

Taking limit as , we get

Or

Thus,

(4) Prove that

Proof:

By Binomial theorem, when , we get

(By definition)

Thus,

(5) Prove that

Proof:

(Using )

Thus,

(6) Prove that

Proof:

We know that

[Dividing throughout by ]

Thus,

Example:

Evaluate:

Solution:

Put, where

Thus,

Developed by: