Question

Evaluate the given limit
$\sf \lim_{x \to 0} (\cos\sec x - \cot x)$
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Answer 1

Correct Question:-

Evaluate the given limit : $\sf\lim_{x \to 0} ( cosec \ \ x - cot \ \ x )$

Given:-

Evaluate the given limit : $\sf\lim_{x \to 0} ( cosec \ \ x - cot \ \ x )$

To Find:-

Evaluate the limit.

Note:-

●》Here, the inverse of cosec x is sin x ; When cot x is expand it becomes $\dfrac{cos \ \ x}{sin \ \ x}$ .

●》Formula of : 1 - cos x = 2 sin² $\dfrac{x}{2}$, sin 2 x = $2 \ \ sin \dfrac{x}{2} × cos \dfrac{x}{2}$ .

●》$\dfrac{sin \frac{x}{2}}{cos \frac{x}{2}} \ \ changes \ \ to \implies tan \dfrac{x}{2}$ .

Solution:-

$\huge\red{\sf\lim_{x \to 0} ( cosec \ \ x - cot \ \ x )}$

$\huge\red{ \ \ \ \ Their \ \ Evaluation = ?}$

☆ According to note first point~

▪︎$\sf\lim_{x \to 0} ( cosec \ \ x - cot \ \ x )$

▪︎$\sf\lim_{x \to 0} ( \dfrac{1}{sin \ \ x} - \dfrac{cos \ \ x}{sin \ \ x} )$

▪︎$\sf\lim_{x \to 0} ( \dfrac{1 - cos \ \ x}{sin \ \ x} )$

☆ According to note second point~

▪︎$\sf\lim_{x \to 0} ( \dfrac{2 sin² \frac{x}{2}}{2 sin \frac{x}{2} × cos \frac{x}{2}} )$

▪︎$\sf\lim_{x \to 0} ( \dfrac{2 sin \frac{x}{2} × sin \frac{x}{2}}{2 sin \frac{x}{2} × cos \frac{x}{2}} )$

☆ After dividing 2 by 2, $sin \dfrac{x}{2} \ \ by \ \ sin \dfrac{x}{2}$ ~

▪︎$\sf\lim_{x \to 0} ( \dfrac{sin \frac{x}{2}}{cos \frac{x}{2}} )$

☆ According to note third point~

▪︎$\sf\lim_{x \to 0} ( tan \dfrac{x}{2} )$

☆ Putting the limit value~

▪︎$( tan \dfrac{0}{2} )$

☆ After dividing 0 by 2~

▪︎$( tan 0 )$

☆ As tan 0 will be 0~

▪︎$0$

$\huge\pink{Their \ \ Evaluation = 0}$

Answer:-

Hence, Evaluation of : $\sf\lim_{x \to 0} ( cosec \ \ x - cot \ \ x ) = 0$.

:)

Answer 2

$\huge\mathtt\pink{A}\mathtt\red{N}\mathtt\blue{S}\mathtt\green{W}\mathtt\purple{E}\mathtt\green{R}$

Solution:-

$\mathtt \red{ lim_{x \rightarrow0 }(cosec \: x \: - cot \: x)}$

If we put the value of X than cosec x will give undefined

Hence changing state of look:-

$\divideontimes \: \: \: \mathtt \blue{ lim_{x \rightarrow0 }( \frac{1}{sin \: x} \: - \frac{cos \: x}{sin \: x} )}$

$\divideontimes \: \: \: \: \mathtt \green{ lim_{x \rightarrow0 }( \frac{1 - cos \: x}{sin \: x})} = \mathtt \orange { lim_{x \rightarrow0 }( \frac{2 { \sin}^{2} \frac{x}{2} }{2sin \frac{x}{2} \cos \frac{x}{2} })}$

$\divideontimes \: \: \: \: \: \ \: \mathtt \pink{ lim_{x \rightarrow0 }( \frac{ \sin \frac{x}{2} }{ \cos \frac{x}{2} })} =$

$\divideontimes \: \: \: \: \: \mathtt \green{ lim_{x \rightarrow0 }( \tan \frac{x}{2} ) }$

$\divideontimes \: \: \: \: \mathtt \red{if \: x \: tending \: to \: zero \: than \: \rightarrow}$

$\divideontimes \: \: \: \: \: \mathtt \blue{ \boxed{ tan \frac{x}{2} = 0} }$