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  • Evaluate    <img alt="\mathrm{\lim _{x \rightarrow 0} \frac{1-\cos (x) \sqrt{\cos (2 x)} \sqrt[3]{\cos (3 x)}}{x^2}}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7

Evaluate    \mathrm{\lim _{x \rightarrow 0} \frac{1-\cos (x) \sqrt{\cos (2 x)} \sqrt[3]{\cos (3 x)}}{x^2}}

Option: 1

0


Option: 2

3


Option: 3

5


Option: 4

1


Answers (1)

best_answer

Note that 

\mathrm{\cos (x)=1-\frac{1}{2} x^2+O\left(x^4\right)}                                       (1)

and      \mathrm{(1-x)^a=1-a x+O\left(x^2\right)}                          (2)

and       \mathrm{(1+a x)(1+b x)=1+(a+b) x+O\left(x^2\right)}      (3)

Thus,

             \begin{aligned} & 1-\cos (x) \cos (2 x)^{1 / 2} \cos (3 x)^{1 / 3} \; \; \; \; \; \; \; \\ \\& =1-\left(1-\frac{1}{2} x^2\right)\left(1-\frac{1}{2} \cdot 4 x^2\right)^{1 / 2}\left(1-\frac{1}{2} \cdot 9 x^2\right)^{1 / 3}+O\left(x^4\right)\; \; \; \; \; \; \; \; \; (4) \\ \\& =1-\left(1-\frac{1}{2} x^2\right)\left(1-x^2\right)\left(1-\frac{3}{2} x^2\right)+O\left(x^4\right)\; \; \; \; \; \; \; \; \; \; (5) \\ \\& =1-\left(1-3 x^2\right)+O\left(x^4\right) \; \; \; \; \; \; \; \; (6)\\ \\& =3 x^2+O\left(x^4\right)\; \; \; \; \; \; \; \; \; (7) \end{aligned}

Explanation:
(4): apply (1)
(5): apply (2)
(6): apply (3)
(7): simplify

Therefore , 

                    \begin{aligned} \mathrm{\lim _{x \rightarrow 0} \frac{1-\cos (x) \cos (2 x)^{1 / 2} \cos (3 x)^{1 / 3}}{x^2}} & =\mathrm{\lim _{x \rightarrow 0} 3+O\left(x^2\right)} \\ & =3 \end{aligned}

Posted by

vishal kumar

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