Baseband representation of a LFM chirp signal
The one element missing from your problem is the time duration of the pulse - I'll assume it is $T$. The general form of a chirp is
$$
p(t)=\exp\left(j2\pi(f_0t+\tfrac{1}{2}\alpha t^2)\right),
$$
where $f_0$ is the frequency at the start of the chirp and $\alpha$ is the chirp rate. For you case $f_0 =-f_{limit}$. To find $\alpha$ we set the end frequency to be $f_{limit}$ and solve
$$ f_0+\alpha T =f_{limit}$$ which gives:
$$ \alpha = \frac{2f_{limit}}{T}$$

Thus the pulse is given by:
$$
p(t) =\exp \left(j2\pi\left(-f_{limit}t+\frac{f_{limit}}{T}t^2\right)\right)
$$