Question

Find the limits of $\frac{e^{mx}-e^{ma}}{x-a},$ when $x=a$.

Answer 1

To find the limits of $\frac{e^{mx}-e^{ma}}{x-a}$ when $x=a$

i.e. $\underset{x\to a}{lim}\frac{e^{mx}-e^{ma}}{x-a}$

$=\underset{x\to a}{lim}\left(e^{ma}\right)\left\{\frac{e^{mx-ma}-1}{x-a}\right\}$

$=\left(m{e}^{ma}\right)\underset{x\to a}{lim}\left\{\frac{e^{m(x-a)}-1}{m(x-a)}\right\}$

$=m{e}^{ma}$ ......... $[\because \underset{x\to 0}{lim}\frac{(a^x-1)}{x}={\log }_{e}a]$