Question

Examine the continuity of the following functions at given points
$f\left(x\right)=\frac{e^{5x}-e^{2x}}{\sin 3x}$, for $x\ne 0$
$=1$, for $x=0$

Answer 1

we will the limit value at $x=0$

$\Rightarrow li{m}_{x\to 0}\frac{e^{5x}-e^{2x}}{\sin 3x}$

it is in the form $\frac{0}{0}$ . use L hopitals rule

$\Rightarrow {lim}_{x\to 0}\frac{5e^{5x}-2e^{2x}}{3\cos 3x}$

now substitute $x=0$

$\Rightarrow \frac{5-2}{3}=1$

$li{m}_{x\to 0}\frac{e^{5x}-e^{2x}}{\sin 3x}=f\left(0\right)=1$

$\therefore$ given function is continuous