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  4. The value of undersetx arrow 0l i m(l n (10 - 9 cos 2 x)/l n2 (sin ⁡ 3 x + 1)) is equal to

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Q. The value of $\underset{x \rightarrow 0}{l i m}\frac{l n \left(10 - 9 cos 2 x\right)}{l n^{2} \left(sin ⁡ 3 x + 1\right)}$ is equal to

NTA AbhyasNTA Abhyas 2020Limits and Derivatives

Solution:

$\underset{x \rightarrow 0}{l i m}\frac{l n \left(10 - 9 cos 2 x\right)}{l n^{2} \left(sin ⁡ 3 x + 1\right)}=\underset{x \rightarrow 0}{l i m}\frac{l n \left\{1 + \left(9 - 9 cos ⁡ 2 x\right)\right\}}{l n^{2} \left(1 + sin ⁡ 3 x\right)}$
$=\underset{x \rightarrow 0}{l i m}\frac{9 \left(1 - cos 2 x\right)}{\left(sin ⁡ 3 x\right)^{2}}=\underset{x \rightarrow 0}{l i m}\frac{9 \left(2 x^{2}\right)}{\left(3 x\right)^{2}}=2$