Question

Evaluate
cosecx-Cotx/x
limit

x_0​

Answer 1

$\begin{gathered}\Large{\bold{\pink{\underline{Formula \: Used \::}}}} \end{gathered}$

$\boxed{ \red{ \: {(1). \: 1 - cos2x = 2 {sin}^{2} x}}}$

$\boxed{ \red{ \: {(2). \:sin2x = 2 \: sinx \: cosx}}}$

$\boxed{ \red{ \: {(3). \:\tt \ \: \tt \:\lim_{x\to0} \:\dfrac{tanx}{x} = 1}}}$

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$\large\underline\purple{\bold{Solution :- }}$

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$\tt \ \: :  ⟼ \tt \:\lim_{x\to0} \:\dfrac{cosecx \: - \: cotx}{x}$

☆ On substituting the value of x directly, we get indeterminant form.

$\tt \ \: :  ⟼ \tt \:\lim_{x\to0} \:\dfrac{\dfrac{1}{sinx} - \dfrac{cosx}{sinx} }{x}$

$\tt \ \: :  ⟼ \tt \:\lim_{x\to0} \:\dfrac{1 - cosx}{x \: \times sinx}$

$\tt \ \: :  ⟼ \tt \:\lim_{x\to0} \:\dfrac{2 \: {sin}^{2}\dfrac{x}{2} }{x \: \times 2 \: sin\dfrac{x}{2} \: cos\dfrac{x}{2} }$

$\tt \ \: :  ⟼ \tt \:\lim_{x\to0} \:\dfrac{tan \: \dfrac{x}{2} }{x}$

$\tt \ \: :  ⟼ \tt \:\lim_{x\to0} \:\dfrac{tan \: \dfrac{x}{2} }{\dfrac{x}{2} \times 2}$

$\tt \ \: :  ⟼ \dfrac{1}{2}$

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