Question

Evaluate: ${lim}_{x\to 0}\frac{a^x+b^x-c^x-d^x}{x}$

Answer 1

$\to \underset{x\to 0}{lim}\frac{a^x+b^x-c^x-d^x}{x}$

this is $\frac{0}{0}$ from

$\therefore =$ Applying $L-$Hospital rule given

$\underset{x\to 0}{lim}(a^{x}lna+b^{x}lnb-c^{x}lnc-d^{x}lnd)$

$=lna+lnb-lnc-lnd$

$=ln\left(\frac{ab}{cd}\right)$

$\therefore \underset{x\to 0}{lim}\frac{a^x+b^x+(-c^x-d^x)}{x}=ln\left(\frac{ab}{cd}\right)$