Equivalence of two expressions

Question

I have the following expression $$\varphi = -5\Delta t + \sqrt{25(\Delta t)^{2}+1}$$ and I want to show that in fact this is equal to $$\varphi = e^{-5\Delta t} + \mathcal{O}(\Delta t^{3}).$$ To do so I was considering the taylor expansion of $e^{x}$, ie. $$e^{x} = 1+x+\frac{x^{2}}{2}+\mathcal{O}(x^{3})$$ and then if we let $x=-5\Delta t$ we have that $$e^{-5\Delta t}=1-5\Delta t + \frac{25(\Delta t)^{2}}{2}+\mathcal{O}(\Delta t^{3})$$ or we can write $$e^{-5\Delta t}+\mathcal{O}(\Delta t^{3})=1-5\Delta t + \frac{25(\Delta t)^{2}}{2}$$ Now I can sort of see a similar relation here but I'm not sure what the next step would be.

Answer 1

Hint: Taylor expand $f(x) = \sqrt{1+25x^{2}}$. You should find $f(0) = 1, f'(0)=0, f''(0) = 25$. Hence...