Question

Evaluate $\underset{x\to 2}{lim}\left(\frac{e^{3x}-1}{x}\right)$

Answer 1

$\underset{x\to 0}{lim}\left(\frac{e^{3x}-1}{x}\right)=\underset{3x\to 0}{lim}\{\left(\frac{e^{3x}-1}{3x}\right)\times 3\}[\because (x\to 0)\Rightarrow (3x\to 0\left)\right]$

$=3\times \underset{y\to 0}{lim}\left(\frac{e^y-1}{y}\right),wherey=3x$

$=(3\times 1)=3[\because \underset{y\to 0}{lim}\left(\frac{e^y-1}{y}\right)=1]$