Question

If $f\left(x\right)=\frac{{\sin }^4\left(\frac{1}{x}\right)-{\sin }^2\left(\frac{1}{x}\right)+1}{{\cos }^4\left(\frac{1}{x}\right)-{\cos }^2\left(\frac{1}{x}\right)+1}$ is to be made continuous at $x=0$, then $f\left(0\right)$ should be equal to

• A. $0$
• B. $1$
• C. $1/3$
• D. $1/2$

Answer 1

The correct option is B $1$

$f\left(x\right)=\frac{{\sin }^4\left(\frac{1}{x}\right)-{\sin }^2\left(\frac{1}{x}\right)+1}{{\cos }^4\left(\frac{1}{x}\right)-{\cos }^2\left(\frac{1}{x}\right)+1}$

$=\frac{{\sin }^2\left(\frac{1}{x}\right)(1-{\cos }^2\left(\frac{1}{x}\right))-{\sin }^2\left(\frac{1}{x}\right)+1}{{\cos }^2\left(\frac{1}{x}\right)(1-{\sin }^2\left(\frac{1}{x}\right))-{\cos }^2\left(\frac{1}{x}\right)+1}$

$=\frac{1-{\sin }^2\left(\frac{1}{x}\right){\cos }^2\left(\frac{1}{x}\right)}{1-{\sin }^2\left(\frac{1}{x}\right){\cos }^2\left(\frac{1}{x}\right)}$

Hence for f to be continuous at $x=0,$ $f\left(0\right)=\underset{x\to 0}{lim}f\left(x\right)=1$