${D_g} = \\left\\{ {x \\in \\mathbb{R}: – \\frac{x}{2} \\in \\mathbb{R}} \\right\\} = \\mathbb{R}$.\nComo $$\\begin{array}{*{20}{l}}
\n{{m_1}}& = &{\\mathop {\\lim }\\limits_{x \\to\u00a0 – \\infty } \\frac{{g(x)}}{x}} \\\\
\n{}& = &{\\mathop {\\lim }\\limits_{x \\to\u00a0 – \\infty } \\frac{{x – 1 + {e^{ – \\frac{x}{2}}}}}{x}} \\\\
\n{}& = &{\\mathop {\\lim }\\limits_{x \\to\u00a0 – \\infty } \\left( {1 – \\frac{1}{x} + \\frac{{{e^{ – \\frac{x}{2}}}}}{x}} \\right)} \\\\
\n{}& = &{1 – \\mathop {\\lim }\\limits_{x \\to\u00a0 – \\infty } \\left( {\\frac{1}{x}} \\right) + \\frac{1}{2}\\mathop {\\lim }\\limits_{x \\to\u00a0 – \\infty } \\left( {\\frac{{{e^{ – \\frac{x}{2}}}}}{{\\frac{x}{2}}}} \\right)} \\\\
\n{}& = &{1 + 0 – \\frac{1}{2}\\underbrace {\\mathop {\\lim }\\limits_{y \\to\u00a0 + \\infty } \\left( {\\frac{{{e^y}}}{y}} \\right)}_{ + \\infty }} \\\\
\n{}& = &{ – \\infty }
\n\\end{array}$$
\nent\u00e3o o gr\u00e1fico da fun\u00e7\u00e3o n\u00e3o possui ass\u00edntota quando $x \\to\u00a0 – \\infty $.<\/p>\n
No entanto, o gr\u00e1fico de $g$ admite como ass\u00edntota ob\u00edqua a reta de equa\u00e7\u00e3o $y=x-1$, quando $x \\to\u00a0 + \\infty $, pois $$\\mathop {\\lim }\\limits_{x \\to\u00a0 + \\infty } \\left[ {g(x) – \\left( {x – 1} \\right)} \\right] = \\mathop {\\lim }\\limits_{x \\to\u00a0 + \\infty } {e^{ – \\frac{x}{2}}} = 0$$
\n\u00ad<\/p>\n<\/li>\n
A fun\u00e7\u00e3o $g$ \u00e9 cont\u00ednua em $\\mathbb{R}$, pois \u00e9 a soma de uma fun\u00e7\u00e3o afim com uma fun\u00e7\u00e3o composta de fun\u00e7\u00f5es cont\u00ednuas tamb\u00e9m em $\\mathbb{R}$. Consequentemente, a fun\u00e7\u00e3o \u00e9 tamb\u00e9m cont\u00ednua no intervalo $\\left[ { – 3, – 2} \\right]$.\nComo $$g( – 3) =\u00a0 – 3 – 1 + {e^{\\frac{3}{2}}} =\u00a0 – 4 + e\\sqrt e\u00a0 > 0$$ e $$g( – 2) =\u00a0 – 2 – 1 + {e^1} =\u00a0 – 3 + e < 0$$ ent\u00e3o $$g( – 2) < 0 < g( – 3)$$<\/p>\n
Logo, de acordo com o Teorema de Bolzano-Cauchy, $\\exists x \\in \\left] { – 3, – 2} \\right[:g(x) = 0$, isto \u00e9, a fun\u00e7\u00e3o $g$ admite pelo menos um zero no intervalo considerado.
\n\u00ad<\/p>\n<\/li>\n<\/ol>\n