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  4. Evaluate displaystyle limx → 0(sin 4x/sin 2x)=

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Q. Evaluate $\displaystyle \lim_{x \to 0}$$\frac{sin\,4x}{sin\,2x}=$

Limits and Derivatives

Solution:

$\displaystyle \lim_{x \to 0}$$\frac{sin\,4x}{sin\,2x}$
$=\displaystyle \lim_{x \to 0}$$\left[\frac{sin\,4x}{4x}\cdot\frac{2x}{sin\,2x}\cdot2\right]$
$=2 \cdot \displaystyle \lim_{x \to 0}$$\left\{\left[\frac{sin\,4x}{4x}\right] \cdot \left[\frac{sin\,2x}{2x}\right]\right\}$
$=2\cdot \displaystyle \lim_{4x \to 0}$$\left[\frac{sin\,4x}{4x}\right] \cdot \displaystyle \lim_{2x \to 0}$$\left[\frac{sin\,2x}{2x}\right]$
$=2 \cdot 1 \cdot 1=2$ (as $x \to 0$, $4x \to 0$ and $2x \to 0$)