Evaluate the following limit : lim(x→0) (2 sin x - sin 2x)/x^3

Question

Evaluate the following limit : \(\lim\limits_{\text x \to0}\cfrac{2\,sin\,\text x-sin\,2\text x}{\text x^3} \)

lim(x→0) (2 sin x - sin 2x)/x³

Answer 1

\(\lim\limits_{\text x \to0}\cfrac{2\,sin\,\text x-sin\,2\text x}{\text x^3} \)

lim(x→0) (2 sin x - sin 2x)/x³

To Find: Limits

NOTE: First Check the form of imit. Used this method if the limit is satisfying any one from 7 indeterminate form.

In this Case, indeterminate Form is \(\cfrac00\) 

We know that sin 2x = 2 sin x cos x

 Formula used : \(\lim\limits_{\text x \to 0}\cfrac{1-cos\,\text x}{\text x^2}=\cfrac12\) and \(\lim\limits_{\text x \to 0}\cfrac{sin\,\text x}{\text x}=1\) 

So, by using the above formula, we have