Question

Let $f\left(x\right)=\left|x\right|$ and $g\left(x\right)=\left[x\right],$ (where, $\left[.\right]$ denotes the greatest integer function). Then, ${\left(fog\right)}^{{}^{\prime }}\left(-1\right)$ is

• A. $0$
• B. does not exist
• C. $-1$
• D. $1$

Answer 1

Answer: (B)

$(f\circ g)(x)=f(g(x))=f([x])=|[x]|$
Now, LHL of $(f\circ g{)}^{\prime }(-1)={\lim }_{h\to 0^{+}}\frac{(f\circ g)(-1-h)-(f\circ g)(-1)}{-h}$
$=\underset{h\to 0^{+}}{\lim }\frac{|\left[-1-h\right]|-|\left[-1\right]|}{-h}$
$=\underset{h\to 0^{+}}{\lim }\frac{|-2|-|-1|}{-h}=\underset{h\to 0^{+}}{\lim }\frac{1}{-h}\to -\in fty$
$\therefore {\left(fog\right)}^{{}^{\prime }}\left(-1\right)$ does not exist.