limit with GIF and FPF

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In this I could not understand how they got te value of a,b and $\lambda$.

Can anybody please explain me .

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Please note that they have written the statement: "Function is diffrentiable in R"

This simply means $$\lim_{x\rightarrow 0^-}f(x)=\lim_{x\rightarrow 0^+}f(x)=f(0)$$ $$1=b=\lambda$$ This was by applying continuity.

By applying differentiation rules across $x=0$ we get: $$a=e^0=1$$ Thus $$a+b+\lambda=3$$