Choosing the parameter in a function to bring the function as close to 1 as possible when x is approximately equal to zero.

Question

The problem states the following:

How should the parameter λ be chosen so that f(x) = e^(-λx)/(1+2sin(x)) remains as close to 1 as possible, when x ≈ 0?

I understand that the solution first simplifies the function into a simpler approximation. However, the last line in the solution states the following: So F is const. to 1st order if λ = -2. I dont quite understand what is meant with this statement.

Problem with Solution

Answer 1

It means that $\lambda$ is chosen so that $(1 - \lambda x) (1 - 2x)$ will equal $1 + (0 \times x) + (\text{something}~ \times x^2).$