We look at how sums of exponential functions of the form \[f(x)=a\cdot g^x+b\cdot h^x, \quad \text{with }g\neq h\] behave for large \(x\).

Comparing the graphs we see that the graph of \(\left(\tfrac{2}{5}\right)^x-\left(\tfrac{1}{5}\right)^x\) resembles the graph of \(\left(\tfrac{2}{5}\right)^x\) for large values of \(x\) .

In other words, the \(x\mapsto\left(\tfrac{2}{5}\right)^x-\left(\tfrac{1}{5}\right)^x\) function is very similar to the \(x\mapsto \left(\tfrac{2}{5}\right)^x\) \(x\) for large values of \(x\).

This can also be seen by extracting one of the exponential functions as a factor.

It follows from \(\left(\tfrac{2}{5}\right)^x-\left(\tfrac{1}{5}\right)^x=\left(\tfrac{2}{5}\right)^x\cdot \left(1-\left(\tfrac{1}{2}\right)^x\right)\) that for large values of \(x\) the expression \(1-\left(\tfrac{1}{2}\right)^x\) between brackets is close to \(1\). Then the function looks like \(x\mapsto \left(\tfrac{2}{5}\right)^x\).