Question

Solve: $\underset{x\to 0}{lim}(cosecx-\cot x)$

Answer 1

${lim}_{x\to 0}(cosecx-cotx)$

$={lim}_{x\to 0}(\frac{1}{sinx}-\frac{cosx}{sinx})$

Using cosec $\theta =\frac{1}{sin\theta }$

$cot\theta =\frac{cos\theta }{sin\theta }$

$={lim}_{x\to 0}\frac{1-cosx}{sinx}$

putting x = 0

$=\frac{1-cos0}{sin0}$

$=\frac{1-1}{0}$

$=\frac{0}{0}$

Hence,we need to simplify

${lim}_{x\to 0}\frac{1-cosx}{sinx}$

$={lim}_{x\to 0}\frac{1-cosx}{sinx}\times \frac{1+cosx}{1+cosx}$

$={lim}_{x\to 0}\frac{1^2-co{s}^{2}x}{sinx(1+cosx)}$

$={lim}_{x\to 0}\frac{1-co{s}^{2}x}{sinx(1+cosx)}$

$={lim}_{x\to 0}\frac{si{n}^{2}x}{sinx(1+cosx)}$

$={lim}_{x\to 0}\frac{sinx}{(1+cosx)}$

Putting x = 0

$=\frac{sin0}{(1+cos0)}$

$=\frac{0}{1+1}$

$=\frac{0}{2}$

$=0$