One of the reasons why these equations are vaild 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 xdT as the power of e. Since taylor series expansion is nearly accurate only when its power is small, xdT should be small and hence this works only for small range of dT.

If you want the complete equation, here it is: L = L0e^(x(T-T0)). Here L0 is length at temp T0 and L is the length at T, x is the coefficient of linear expansion. So the equation you are using comes from approximating this above equation when xdT is small.

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.