Question
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow{\frac{\pi}{6}}}\frac{\cot^2\text{x}-3}{\text{cosec x}-2}$
$\lim\limits_{\text{x}\rightarrow{\frac{\pi}{6}}}\frac{\cot^2\text{x}-3}{\text{cosec x}-2}$
✓
Answer
$\lim\limits_{\text{x}\rightarrow{\frac{\pi}{6}}}\frac{\cot^2\text{x}-3}{\text{cosec x}-2}$
$=\lim\limits_{\text{x}\rightarrow{\frac{\pi}{6}}}\frac{\big(\text{cosec}^2\text{x}-1\big)-3}{\text{cosec x}-2}$
$=\lim\limits_{\text{x}\rightarrow{\frac{\pi}{6}}}\frac{\big(\text{cosec}^2\text{x}-4\big)}{\text{cosec x}-2}$
$=\lim\limits_{\text{x}\rightarrow{\frac{\pi}{6}}}\frac{\big(\text{cosec }\text{x}-2\big)\big(\text{cosec }\text{x}+2\big)}{\text{cosec x}-2}$
$=\lim\limits_{\text{x}\rightarrow{\frac{\pi}{6}}}\text{cosec }\text{x}+2$
$=\text{cosec}\frac{\pi}{6}+2$
$=2+2$
$=4$
$=\lim\limits_{\text{x}\rightarrow{\frac{\pi}{6}}}\frac{\big(\text{cosec}^2\text{x}-1\big)-3}{\text{cosec x}-2}$
$=\lim\limits_{\text{x}\rightarrow{\frac{\pi}{6}}}\frac{\big(\text{cosec}^2\text{x}-4\big)}{\text{cosec x}-2}$
$=\lim\limits_{\text{x}\rightarrow{\frac{\pi}{6}}}\frac{\big(\text{cosec }\text{x}-2\big)\big(\text{cosec }\text{x}+2\big)}{\text{cosec x}-2}$
$=\lim\limits_{\text{x}\rightarrow{\frac{\pi}{6}}}\text{cosec }\text{x}+2$
$=\text{cosec}\frac{\pi}{6}+2$
$=2+2$
$=4$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Differentiate the following from first principle:
→$\frac{\text{x}^2+1}{\text{x}}$
Find three numbers in G.P. whose Product is 729 and the sum of their products in pairs is 819.
The 4th and 7th terms of G.P. are $\frac{1}{27}\text{ and }\frac{1}{729}$ respectively. Find the sum of n terms of the G.P.
→If than $\alpha=\text{x}+1,\tan\beta=\text{x}-1,$ prove that $2\cot(\alpha-\beta)=\text{x}^2$
→The ratio of the A.M. and G.M. of two positive numbers a and b is 
Show that $a : b = \left( \begin{array} { c } { m + \sqrt { m ^ { 2 } - n ^ { 2 } } } \end{array} \right) : \left( m - \sqrt { m ^ { 2 } - n ^ { 2 } } \right)$
→ Show that $a : b = \left( \begin{array} { c } { m + \sqrt { m ^ { 2 } - n ^ { 2 } } } \end{array} \right) : \left( m - \sqrt { m ^ { 2 } - n ^ { 2 } } \right)$Find the term independent of x in the expansion of the following expressions:
$\Big(\frac{3\text{x}^{2}}{2}-\frac{1}{3\text{x}}\Big)^{9}$
Prove that the number of subsets of a set containing n distinct elements is 2n for all $\text{n}\in\text{N}.$
→Evaluate the following limits.
$\lim\limits_{\text{x} \rightarrow 0}\frac{(\sin(\alpha+\beta)\text{x}+\sin(\alpha-\beta)\text{x}+\sin2\alpha\cdot\text{x}}{\cos2\beta\text{x}-\cos2\alpha\text{x}}$
→$\lim\limits_{\text{x} \rightarrow 0}\frac{(\sin(\alpha+\beta)\text{x}+\sin(\alpha-\beta)\text{x}+\sin2\alpha\cdot\text{x}}{\cos2\beta\text{x}-\cos2\alpha\text{x}}$
If $\text{a}=\cos\theta+\text{i}\sin\theta,$ find the value of $\frac{1+\text{a}}{1-\text{a}}.$
→Solve the following equations:
$2^{\sin^2\text{x}}+2\cos^{2\text{x}}=2\sqrt{2}$
→$2^{\sin^2\text{x}}+2\cos^{2\text{x}}=2\sqrt{2}$