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limits of Trigonometric Function Solution and Examples  (with 10+ examples)

limits of Trigonometric Function Solution and Examples (with 10+ examples)

05/09/2021
limits of Trigonometric Function| Theorem and Examples (limit of trig function)
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TRIGONOMETRIC LIMIT

Limits involving trigonometric functions are defined as trigonometric limits. To study the trigonometric limit, we study the following trigonometric transformation and theorem

Sandwich theorem

 Let us consider any three function

such that

 

 

Then sandwich theorem can be defined as the limit function such that

 

Proof: since the last two functions are equal i.e.

 and also we have

 or

 Which indicate that

 hence by intuition, we can write

 

Important Theorems 

1. Theorem-1

 

Proof; to prove this relation let us consider a circle having a unit radius and center at the origin.

From figure

∠AOP=θ 

 

 

Now we can write,

0≤sinθ≤θ

As o0 then θ→0

By using the sandwich theorem, we can write

 

 

 

2. Theorem-2

   

Proof:  To prove the above relation, let us consider a circle having radius r and center 0

let us consider any point p on the circle then OP=OA=r

Let θ be the angle subtended by the arc AP with center O. let us draw perpendicular PN on OA and draw tangent PA and PQ at point P.

here we have

 △OPA≤  area of sector OPA ≤ are of the △OPQ        … (1)

Now,

Area of the  

 

        … (2) 

Now we know that

Angle 2πC=Area πr2 

       … (3)

Again, are of the triangle

         … (4)

From equation (1), (2), (3), and (4) we get

△OPA≤  are of sector OPA ≤  are of the △OPQ  

     

sinθθ≤  tanθ

 

 

 

Taking limit both side

 

 

By using the sandwich theorem, we can write

 proved.

 

Theorem-3

 

Proof:

 

      

4. Theorem-4

 

Proof:

 

 

SOLVED EXAMPLE

Exampel-1.

Evaluate

Solution:

 

 

 

Example-2  Evaluate

Solution:

 Ans.

 

Example-3 evaluate

Solution:

 

  

 ∵limθ→0sinθθ=1

 Ans.

 

Example-4  

 

 

 

 Ans.

 

Example-5 Evaluate   

Solution:

 

where the angle is in radian measure of the system ( 1800=πc)   

 

 Ans.

Example-6. Evaluate

Solution:  when we put x=a then given function yield  form which is indeterminant form

 

    Ans.

Example-7 Evaluate

Solution: when we put x=θ then the above function yields     form which is indeterminant form.

 

 

 

 

 

 

 

Ans.  

Exercise 

1. What is the limit of trigonometry function? 

2. what is the sandwich theorem? define in detail and prove the sandwich theorem.

3. Prove four important theorem needed to solve the limit involving trigonometric function.

4. evaluate

5. Evaluate  

6. Evaluate 

7.  Prove geometrically relation  

MCQs (answer are encircled)

1. Limit of  

 aseca(tana+1)

aseca

aseca(cota+2)

 cota(sin2a+1)

2. limit of 

 p/q

q/p

q2/p2

 p2/q2

3. limit of  

 1

3

2

 0

4.The relation  

  True 

 False 

 inderrminant 

 True and accordance with sandwich theorem.

Want to study more topics?

 

 





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