Question

$\underset{x\to 0}{lim}\left[\ln \right(1+{\sin }^{2}x\left)\cot {\ln }^2\right(1+x\left)\right]=$

• A. $-1$
• B. $1$
• C. $0$
• D. $2$

Answer 1

Answer: (B)

$1$

$\underset{x\to 0}{lim}\left[\ln \right(1+{\sin }^{2}x\left)\cot {\ln }^2\right(1+x\left)\right]=\underset{x\to 0}{lim}\frac{\ln (1+{\sin }^{2}x)}{{\sin }^{2}x}\cdot \frac{{\sin }^{2}x}{x^2}\cdot \frac{{\ln }^2(1+x)}{\tan {\ln }^2(1+x)}\cdot \frac{x^2}{{\ln }^2(1+x)}$

$=\underset{x\to 0}{lim}\frac{\ln (1+{\sin }^{2}x)}{{\sin }^{2}x}\cdot {\left(\frac{\sin x}{x}\right)}^2\cdot \frac{{\ln }^2(1+x)}{\tan {\ln }^2(1+x)}\cdot {\left(\frac{x}{\ln (1+x)}\right)}^2=1\cdot 1\cdot 1\cdot 1=1$