Question

Left hand derivative and right hand derivative of a function $f ( x )$ at a point $x = a$ are defined as
$f^{\prime}\left(a^{-}\right)=\displaystyle\lim _{h \rightarrow 0^{+}} \frac{f(a)-f(a-h)}{h}=\displaystyle\lim _{h \rightarrow 0^{-}} \frac{f(a+h)-f(a)}{h} \text { and } $
$f^{\prime}\left(a^{+}\right)=\displaystyle\lim _{h \rightarrow 0^{+}} \frac{f(a+h)-f(a)}{h}=\lim _{h \rightarrow 0^{-}} \frac{f(a)-f(a-h)}{h}=\displaystyle\lim _{x \rightarrow a^{+}} \frac{f(a)-f(x)}{a-x} $ respectively
Let $f$ be a twice differentiable function. We also know that derivative of an even function is odd function and derivative of an odd function is even function.
If $f$ is odd, which of the following is Left hand derivative of $f$ at $x=-a$

• A. $\displaystyle\lim _{h \rightarrow 0^{-}} \frac{f(a-h)-f(a)}{-h}$
• B. $\displaystyle\lim _{h \rightarrow 0^{-}} \frac{f(h-a)-f(a)}{h}$
• C. $\displaystyle\lim _{h \rightarrow 0^{+}} \frac{f(a)+f(a-h)}{-h}$
• D. $\displaystyle\lim _{h \rightarrow 0^{-}} \frac{f(-a)-f(-a-h)}{-h}$

Answer 1

Answer: (A)

$L H D=\displaystyle\lim _{h \rightarrow 0^{-}} \frac{f(-a+h)-f(-a)}{h}$
$=\displaystyle\lim _{h \rightarrow 0^{-}} \frac{-f(a-h)+f(a)}{h}=\displaystyle\lim _{h \rightarrow 0^{-}} \frac{f(a-h)-f(a)}{-h}$