Question

${lim}_{x\to 0}\frac{ta{n}^{-1}x-si{n}^{-1}x}{x^3}\text{is equal to}$

• A. 1/2
• B. -1/2
• C. 1
• D. -1

Answer 1

The correct option is B

-1/2

$\underset{x\to 0}{lim}\frac{ta{n}^{-1}x-si{n}^{-1}x}{x^3}\left(\frac{0}{0}form\right)\underset{x\to 0}{lim}\frac{\frac{1}{1+x^2}-\frac{1}{\sqrt{1-x^2}}}{3x^2}\underset{x\to 0}{lim}\frac{\sqrt{1-x^2}-(1+x^2)}{3x^2(1+x^2)\sqrt{1-x^2}}\underset{x\to 0}{lim}\frac{(1-x^2)-(1+x^2{)}^2}{3x^2(1+x^2)\sqrt{1-x^2}[\sqrt{1-x^2}+(1+x^2\left)\right]}=-\frac{3}{6}=-\frac{1}{2}$