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

**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,

âˆµ *0*≤sinθ≤θ

As *o*→*0 **t**h**en** *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

â–³OPA≤ area of sector OPA ≤ are of the â–³OPQ … (1)

Now,

Area of the

… (2)

Now we know that

*Angle 2*π^{C}*=Area *πr^{2}

… (3)

Again, are of the triangle

… (4)

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

â–³OPA≤ are of sector OPA ≤ are of the â–³OPQ

*sinθ* ≤ *θ*≤ 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 E**valuate

Solution:

Ans.

**Example-3 **evaluate

**Solution:**

âˆµ*lim*θ→0*sin*θθ*=1*** **

Ans.

**Example-4 **

Ans.

**Example-5** Evaluate

**Solution:**

where the angle is in radian measure of the system (âˆµ *180**0**=*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

q^{2}/p^{2}

p^{2}/q^{2}

^{3. limit of }

1

3

2

0

4.The relation * *

True

False

inderrminant

True and accordance with sandwich theorem.

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