Question

lim x → 0 cot 4x / cosec 3x

Answer 1

see attachment....hope it will help u

Answer 2

Answer:

The value if given limit is

$L = lim_{x - > 0} \frac{cot4x}{cosec3x} = \frac{3}{4}$

Step-by-step explanation:

The given limit is

$L = lim_{x - > 0} \frac{cot4x}{cosec3x}$

We know that,

$cot4x = \frac{1}{tan4x} \: and \: cosec3x = \frac{1}{sin3x}$

Therefore the given limit transforms as follows

$L = lim_{x - > 0} \frac{ \frac{1}{tan4x} }{ \frac{1}{sin3x} } \\ = lim_{x - > 0} \frac{sin3x}{tan4x}$

We can rewrite the limit as follows

$L = lim_{x - > 0} \frac{ \frac{sin3x}{3x} }{ \frac{tan4x}{4x} } \times \frac{3}{4 } \\ = \frac{3}{4} \frac{lim_{x - > 0} \frac{sin3x}{3x} }{lim_{x - > 0} \frac{tan4x}{4x} }$

In limits we know two fundamental relations.They are

$lim_{x - > 0} \frac{sin(nx)}{nx} = 1 \\ lim_{x - > 0} \frac{tan(nx)}{nx} = 1$

Therefore,the limit becomes

$L = \frac{3}{4} \times \frac{1}{1} = \frac{3}{4}$

Hence,the value of given limit is

$L = lim_{x - > 0} \frac{cot4x}{cosec3x} = \frac{3}{4}$

#SPJ3