Question

sin3x/sin x. limx=0​

Answer 1

Question :–

$\\ \displaystyle\sf \: \: solve : \lim_{x\:\rightarrow\:0} \left[ \frac { \sin(3x) } { \sin(x) } \right] \:$

ANSWER :–

$\\ \longrightarrow { \pink { \displaystyle\sf \: \: \lim_{x\:\rightarrow\:0} \left[ \frac { \sin(3x) } { \sin(x) } \right] = 3 \:}}$

EXPLANATION :–

$\\ \displaystyle\sf \: \: = \lim_{x\:\rightarrow\:0} \left[ \frac { \sin(3x) } { \sin(x) } \right] \:$

• We should write this as –

$\\ \displaystyle\sf \: \: = \lim_{x\:\rightarrow\:0} \left[ \frac { \sin(3x) } { 3x } \times (3x) \times \dfrac{x}{ \sin(x) } \times \dfrac{1}{x} \right] \:$

• We know that –

$\\ \displaystyle\sf \implies { \red{ \boxed{ \lim_{x\:\rightarrow\:0} \left[ \frac { \sin(x) } { x } \right] = 1}}}\:$

• So that –

$\\ \displaystyle\sf \: \: = \lim_{x\:\rightarrow\:0} \left[ 1 \times 3 \cancel x \times 1 \times \dfrac{1}{ \cancel x} \right] \:$

$\\ \displaystyle\sf \: \: = 3$

Hence , $\displaystyle\sf \: \: \lim_{x\:\rightarrow\:0} \left[ \frac { \sin(3x) } { \sin(x) } \right] = 3 \:$

$\\ \rule{220}{2}$

IMPORTANT FORMULA :–

$\\ \displaystyle\sf \: \: (1) \: \: \lim_{x\:\rightarrow\:0} \left[ \frac{ \sin(x) }{ x} \right] \: = 1$

$\\ \displaystyle\sf \: \: (2) \: \: \lim_{x\:\rightarrow\:0} \left[ \frac{ \tan(x) }{ x} \right] \: = 1$

$\\ \displaystyle\sf \: \: (3) \: \: \lim_{x\:\rightarrow\:0} \left[ \frac{ 1 - { \cos} (x) }{ x {}^{2} } \right] \: = \frac{1}{2}$

$\\ \rule{220}{2}$