Teaching Dirac delta "function" $\delta(t)$ I am about to teach applied mathematics for engineering. So I will teach how to use Laplace Transform to solve differential equations. Some of these differential equations involve the Dirac delta "function" as a forcing term, for example:
$$y''(t)+y'(t)-y(t)=3\delta(t-1).$$
The students have as prerequisite: Calculus II and Calculus III. I don't know how to introduce the Dirac delta function:
1) Should I first talk about the distributions as linear functionals acting on test functions. Then define Dirac delta function as $\langle\,\delta,\varphi\rangle=\varphi(0)$.
or
2) Define $\delta$ as "something" that has a Laplace Transform $\mathcal{L(\delta)}=1$,  that is $\delta$ is a limit (in some sense) of a sequence of functions $f_n$ having a tall spike at the origin with $\lim_{n\to \infty}\mathcal{L(f_n)}=1$. That object can be viewed as a "function" defined by $$\delta(t)=\begin{cases}
0, & t\neq0\\
\infty, & t=0
\end{cases}$$

Some might say that you could pass it off as a "function" which is infinite at $t=0$ and is zero everywhere, with integral equal to $1$. However, the distribution approach might be too complicated, especially for engineering students fresh out of Calculus.
The second approach might be the best. You hide the overly complicated details, while still keeping the spirit of the Dirac delta as something that isn't a proper function. Don't leave out the fact that $\langle f,\delta \rangle = f(0)$ though.
If anything, you can tell the more curious students to seek you later for more details, if you have the time. Not everyone in engineering is interested in mathematical formalism; as long as the tool serves its purpose, it might be fine to them.
