I know that the empirically derived equations$$ \begin{align} \mathrm{d}L & = x \, {L}_{0} \, \mathrm{d}T \tag{1} \\[5px] \mathrm{d}V & = y \, {V}_{0} \, \mathrm{d}T \,, \tag{2} \end{align} $$where $x$ and $y$ are coefficients, describe the relationship between the change in length, $\mathrm{d}L ,$ and the change in volume, $\mathrm{d}V ,$ of a material.
Question: Why are these equations valid only if $\mathrm{d}T$ has a small value?
One of the reasons why these equations are valid for small range of temperature is that these equations derived from approximating the original equations.
The original/complete equation involves an exponential function. For getting to these equations, we use the Taylor series expansion of $e^t ,$ where $t$ is very small, here instead of $t,$ we have $x \, \mathrm{d}T$ as the power of $e.$ Since Taylor series expansion is nearly accurate only when its power is small, $x \, \mathrm{d}T$ should be small and hence this works only for small range of $\mathrm{d}T.$
If you want the complete equation, here it is:$$ L ~=~ {L}_{0} \, {e}^{x \left( T - {T}_{0} \right)} \, . $$Here,
${L}_{0}$ is length at temperature ${T}_{0} ;$
$L$ is the length at $T ;$ and
$x$ is the coefficient of linear expansion.
So, the equation you are using comes from approximating this above equation when $x \, \mathrm{d}T$ is small.
