Get the solution to "Evaluate and simplify: limx→0sin⁡2x3x So, by direct substitution,..."

Duncan Reed

Answered question

2022-01-23

Evaluate and simplify:

lim_{x \to 0} \frac{\sin 2 x}{3 x}

So, by direct substitution, it is indeterminate. I got the derivative of this functio,n and now Im

Answer & Explanation

Lily Thornton

Beginner2022-01-24Added 19 answers

lim_{x \to 0} \frac{\sin 2 x}{3 x} =

= lim_{x \to 0} \frac{23}{\cdot} \frac{\sin 2 x}{2 x} =

= \frac{2}{3} \cdot lim_{x \to 0} \frac{\sin 2 x}{2 x}

= \frac{2}{3}

saennwegoyk

Beginner2022-01-25Added 7 answers

Another way is to use the identity
$\sin 2 x = 2 \sin x \cos x$
whence
$lim_{x \to 0} \frac{\sin 2 x}{3 x} = lim_{x \to 0} \frac{2 \sin x \cos x}{3 x} = \frac{23}{lim_{x \to 0} \frac{\sin x}{x}} (lim_{x \to 0} \cos x) = \frac{23}{x} 1 \times 1 = \frac{23}{}$
where we used the limit laws in conjunction with the well-known fact that
$lim_{x \to 0} \frac{\sin x}{x} = 1$
as well as the fact that cosine is a continuous function.

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