Question

The function $f\left(x\right)=\frac{1+\sin x-\cos x}{1-\sin x-\cos x}$ is not defined at $x=0$. The value of $f\left(0\right)$ so that $f\left(x\right)$ is continuous at $x=0$ is

• A. $1$
• B. $-1$
• C. $0$
• D. $2$

Answer 1

The correct option is B $-1$

Since, on applying the limit, the function is of $\frac{0}{0}$ form,
we apply the L-Hospital's Rule and differentiate both the numerator and the denominator individually.
Thus, the question transforms to,
$\underset{x\to 0}{lim}\frac{cosx+sinx}{sinx-cosx}$

Now applying the limit, we get
$\Rightarrow \frac{0+1}{0-1}=-1$
Thus, for the function to be continuous at $x=0$, $f\left(0\right)=-1$