Question Number 212405 by mathlove last updated on 13/Oct/24
$$\underset{{x}\rightarrow\mathrm{3}} {\mathrm{lim}}\frac{{e}^{{x}} −{e}^{\mathrm{3}} }{{x}−\mathrm{3}}=? \\ $$
Answered by som(math1967) last updated on 13/Oct/24
![lim_(x→3) ((e^3 (e^(x−3) −1))/((x−3))) =e^3 lim_(a→0) ((e^a −1)/e^a ) [let x−3=a ∵x→3∴(x−3)→0 ∴a→0] =e^3 ×1=e^3](Q212407.png)
$$\underset{{x}\rightarrow\mathrm{3}} {\:{lim}}\frac{{e}^{\mathrm{3}} \left({e}^{{x}−\mathrm{3}} −\mathrm{1}\right)}{\left({x}−\mathrm{3}\right)} \\ $$$$={e}^{\mathrm{3}} \:\underset{{a}\rightarrow\mathrm{0}} {{lim}}\frac{{e}^{{a}} −\mathrm{1}}{{e}^{{a}} }\:\:\left[{let}\:{x}−\mathrm{3}={a}\right. \\ $$$$\left.\:\:\:\:\:\:\because{x}\rightarrow\mathrm{3}\therefore\left({x}−\mathrm{3}\right)\rightarrow\mathrm{0}\:\therefore{a}\rightarrow\mathrm{0}\right] \\ $$$$={e}^{\mathrm{3}} ×\mathrm{1}={e}^{\mathrm{3}} \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$
Commented by mathlove last updated on 13/Oct/24
$${thanks}\:{sir} \\ $$