The definition of the derivative is $$g'(a)=\lim \limits_{\delta \rightarrow 0} \frac{g(a+\delta) - g(a)}{\delta}$$
also the left derivative is $$ \lim \limits_{\delta \rightarrow 0^-} \frac{g(a+\delta) - g(a)}{\delta} \tag{*}$$
and the right derivative is $$ \lim \limits_{\delta \rightarrow 0^+} \frac{g(a+\delta) - g(a)}{\delta} \tag{**}$$
but am I right to think that the left derivative is equivalent to $$ \lim \limits_{\delta \rightarrow 0} \frac{g(a-\delta) - g(a)}{-\delta} = \lim \limits_{\delta \rightarrow 0} \frac{g(a) -g(a-\delta)}{\delta}$$ and the right equivalent to just $$ \lim \limits_{\delta \rightarrow 0} \frac{g(a+\delta) - g(a)}{\delta} $$ if this is not correct then what is the correct equivalent definition of (*) and (**) i.e I want to not have to deal with $0^+$ and want a definition interms of just limit $0$