Question

Lt x->pi/4 sin 2x/x​

Answer 1

$\displaystyle \lim_{x\to\dfrac{\pi}{4}}\dfrac{sin2x}{x}=\dfrac{4}{\pi}$

Tips:

We must know the rule,

$\displaystyle \lim_{x\to a}[f(x)\:g(x)]=\lim_{x\to a}f(x)\times \lim_{x\to a}g(x)$

Also, $\displaystyle sin2A=2\:sinA\:cosA$

Step-by-step explanation:

Method 1.

Now, $\displaystyle \lim_{x\to\dfrac{\pi}{4}}\dfrac{sin2x}{x}$

$\displaystyle =\dfrac{\displaystyle\lim_{x\to\dfrac{\pi}{4}}(sin2x)}{\displaystyle\lim_{x\to\dfrac{\pi}{4}}(x)}$

$\displaystyle =\dfrac{sin(2\times \dfrac{\pi}{4})}{\dfrac{\pi}{4}}$

$\displaystyle =\dfrac{sin\dfrac{\pi}{2}}{\dfrac{\pi}{4}}$

$\displaystyle =\dfrac{1}{\dfrac{\pi}{4}}$

$\displaystyle =\dfrac{4}{\pi}$

Method 2.

Now, $\displaystyle \lim_{x\to\dfrac{\pi}{4}}\dfrac{sin2x}{x}$

$\displaystyle =\lim_{x\to\dfrac{\pi}{4}}\dfrac{2\:sinx\:cosx}{x}$

$\displaystyle =2\times\dfrac{\displaystyle \lim_{x\to\dfrac{\pi}{4}}(sinx)\times \displaystyle \lim_{x\to\dfrac{\pi}{4}}(cosx)}{\displaystyle \lim_{x\to\dfrac{\pi}{4}}(x)}$

$\displaystyle =2\times\dfrac{sin\dfrac{\pi}{4}\times cos\dfrac{\pi}{4}}{\dfrac{\pi}{4}}$

$\displaystyle =2\times\dfrac{\dfrac{1}{\sqrt{2}}\times\dfrac{1}{\sqrt{2}}}{\dfrac{\pi}{4}}$

$\displaystyle =2\times\dfrac{\dfrac{1}{2}}{\dfrac{\pi}{4}}$

$\displaystyle =\dfrac{4}{\pi}$

#SPJ3