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, $\left(f o g\right)^{'} \left(- 1\right)$ is

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

Answer 1

Answer: (B)

$(f o g)(x)=f(g(x))=f([x])=|[x]|$
Now, LHL of $(f o g)^{\prime}(-1)=\lim _{h \rightarrow 0^{+}} \frac{(\text { fog })(-1-h)-(\text { fog })(-1)}{-h}$
$=\lim _{h \rightarrow 0^{+}} \frac{|[-1-h]|-\mid[-||1}{-h}$
$=\lim _{h \rightarrow 0^{+}} \frac{|-2|-|-1|}{-h}=\lim _{h \rightarrow 0^{+}} \frac{1}{-h} \rightarrow-\infty$
$\therefore (f o g)^{\prime}(-1)$ does not exist.