Find $(g \circ f)(x)$.

Given that $\mathop {\lim }\limits_{x \to  + \infty } (g \circ f)(x) =  - 3$, find the value of $b$.

attempt to form composite     (M1)

eg$\,\,\,\,\,$$g(1 + {{\text{e}}^{ - x}})$

correct function     A1     N2

eg$\,\,\,\,\,$$(g \circ f)(x) = 2 + b + 2{{\text{e}}^{ - x}},{\text{ }}2(1 + {{\text{e}}^{ - x}}) + b$

[2 marks]

evidence of $\mathop {\lim }\limits_{x \to \infty } (2 + b + 2{{\text{e}}^{ - x}}) = 2 + b + \mathop {\lim }\limits_{x \to \infty } (2{{\text{e}}^{ - x}})$     (M1)

eg$\,\,\,\,\,$$2 + b + 2{{\text{e}}^{ - \infty }}$, graph with horizontal asymptote when $x \to \infty$

Note:     Award M0 if candidate clearly has incorrect limit, such as $x \to 0,{\text{ }}{{\text{e}}^\infty },{\text{ }}2{{\text{e}}^0}$.

evidence that ${{\text{e}}^{ - x}} \to 0$ (seen anywhere)     (A1)

eg$\,\,\,\,\,$$\mathop {\lim }\limits_{x \to \infty } ({{\text{e}}^{ - x}}) = 0,{\text{ }}1 + {{\text{e}}^{ - x}} \to 1,{\text{ }}2(1) + b =  - 3,{\text{ }}{{\text{e}}^{{\text{large negative number}}}} \to 0$, graph of $y = {{\text{e}}^{ - x}}$ or

$y = 2{{\text{e}}^{ - x}}$ with asymptote $y = 0$, graph of composite function with asymptote $y =  - 3$

correct working     (A1)

eg$\,\,\,\,\,$$2 + b =  - 3$

$b =  - 5$     A1     N2

[4 marks]