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Limits and Derivatives — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsLimits and Derivatives1 Mark
MCQ
$\lim\limits_{\text{x} \rightarrow0}\frac{\text{cosec}-\cot\text{x}}{\text{x}}$ is equal to:
  • A
    $-\frac{1}{2}$ 
  • B
    $1$ 
  • ✓
    $\frac{1}{2}$ 
  • D
    $-1$
✓

Answer

Correct option: C.
$\frac{1}{2}$ 
Given $​​\lim\limits_{\text{x} \rightarrow 0}\frac{\text{cosec}\text{x}-\cot\text{x}}{\text{x}}$
$=\lim\limits_{\text{x} \rightarrow 0}\frac{\frac{1}{\sin\text{x}}-\frac{\cos\text{x}}{\sin\text{x}}}{\text{x}}$ 
$ =\lim\limits_{\text{x} \rightarrow 0}\frac{1-\cos\text{x}}{\text{x}\sin\text{x}}$
$=\frac{2\sin^{2}\frac{\text{x}}{2}}{\text{x}\cdot\sin\frac{\text{x}}{2}\cos\frac{\text{x}}{2}}$
$=\lim\limits_{\text{x} \rightarrow1}\frac{\sin\frac{\text{x}}{2}}{\text{x}\cos\frac{\text{x}}{2}} $
$=\frac{\tan\frac{\text{x}}{2}}{\text{x}}$
$=\lim\limits_{\text{x} \rightarrow 0}\frac{\tan\frac{\text{x}}{2}}{2\times\frac{\text{x}}{2}}$
$=\frac{1}{2}\times1$
$=\frac{1}{2}$

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