Exponential over sine

Statement

Calculate $\displaystyle\lim_{x\to 0}\frac{e^{2x}-1}{\sin(3x)}$.

Solution

A $0/0$ form. We use the equivalent functions for $x\to 0$: $e^{\square}-1\sim\square$ with $\square=2x$, and $\sin(\square)\sim\square$ with $\square=3x$. Hence $\frac{e^{2x}-1}{\sin(3x)} \sim \frac{2x}{3x} = \frac{2}{3}.$ Therefore $\lim_{x\to 0}\frac{e^{2x}-1}{\sin(3x)} = \boxed{\frac{2}{3}}.$ (Equivalently, by multiplying and dividing by $2x$ and $3x$ to bring out the standard limits $\frac{e^{2x}-1}{2x}\to 1$ and $\frac{\sin 3x}{3x}\to 1$.)

Links

Topics: Limits
Concepts: Equivalent functions · Standard limits
Skills: Calculating limits
Exercise type: Limit calculation