Ass\u00edntotas<\/strong><\/li>\n<\/ul>\nA fun\u00e7\u00e3o \u00e9 cont\u00ednua em $\\mathbb{R}$, logo o seu gr\u00e1fico n\u00e3o tem ass\u00edntotas verticais.<\/p>\n
$$\\mathop {\\lim }\\limits_{x \\to\u00a0 + \\infty } f(x) = \\mathop {\\lim }\\limits_{x \\to\u00a0 + \\infty } \\frac{{{e^x} – {e^{ – x}}}}{2} = \\frac{1}{2} \\times \\left( {\\underbrace {\\mathop {\\lim }\\limits_{x \\to\u00a0 + \\infty } {e^x}}_{ + \\infty } – \\underbrace {\\mathop {\\lim }\\limits_{x \\to\u00a0 + \\infty } {e^{ – x}}}_0} \\right) =\u00a0 + \\infty $$<\/p>\n
Como a fun\u00e7\u00e3o \u00e9 \u00edmpar, ent\u00e3o $\\mathop {\\lim }\\limits_{x \\to\u00a0 – \\infty } f(x) =\u00a0 – \\infty $.<\/p>\n
Logo, o gr\u00e1fico da fun\u00e7\u00e3o n\u00e3o tem ass\u00edntotas horizontais.<\/p>\n
$$m = \\mathop {\\lim }\\limits_{x \\to\u00a0 + \\infty } \\frac{{f(x)}}{x} = \\mathop {\\lim }\\limits_{x \\to\u00a0 + \\infty } \\frac{{{e^x} – {e^{ – x}}}}{{2x}} = \\frac{1}{2} \\times \\mathop {\\lim }\\limits_{x \\to\u00a0 + \\infty } \\frac{{{e^x}\\left( {1 – {e^{ – 2x}}} \\right)}}{x} = \\frac{1}{2} \\times \\underbrace {\\mathop {\\lim }\\limits_{x \\to\u00a0 + \\infty } \\frac{{{e^x}}}{x}}_{ + \\infty } \\times \\underbrace {\\mathop {\\lim }\\limits_{x \\to\u00a0 + \\infty } \\left( {1 – {e^{ – 2x}}} \\right)}_1 =\u00a0 + \\infty $$<\/p>\n
Portanto, o gr\u00e1fico da fun\u00e7\u00e3o tamb\u00e9m n\u00e3o tem ass\u00edntotas obl\u00edquas.<\/p>\n
\n- Monotonia e extremos<\/strong><\/li>\n<\/ul>\n
Ora, \\[f’\\left( x \\right) = {\\left( {\\frac{{{e^x} – {e^{ – x}}}}{2}} \\right)^\\prime } = \\frac{{{e^x} + {e^{ – x}}}}{2}\\]<\/p>\n
Como $f'(x) > 0,\\forall x \\in {D_f}$, ent\u00e3o a fun\u00e7\u00e3o \u00e9 estritamente crescente no seu dom\u00ednio.<\/p>\n
\\[f”\\left( x \\right) = {\\left( {\\frac{{{e^x} + {e^{ – x}}}}{2}} \\right)^\\prime } = \\frac{{{e^x} – {e^{ – x}}}}{2} = \\frac{{{e^x}\\left( {1 – {e^{ – 2x}}} \\right)}}{2} = f(x)\\]<\/p>\n<\/p>\n
\n\n\n| $x$<\/td>\n | ${ – \\infty }$<\/td>\n | $0$<\/td>\n | \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 ${ + \\infty }$<\/td>\n<\/tr>\n |
\n| $\\frac{{{e^x}}}{2}$<\/td>\n | $+$<\/td>\n | $+$<\/td>\n | $+$<\/td>\n<\/tr>\n |
\n| ${1 – {e^{ – 2x}}}$<\/td>\n | $-$<\/td>\n | $0$<\/td>\n | $+$<\/td>\n<\/tr>\n |
\n| $f”(x) = \\frac{{{e^x}\\left( {1 – {e^{ – 2x}}} \\right)}}{2}$<\/td>\n | $-$<\/td>\n | $0$<\/td>\n | $+$<\/td>\n<\/tr>\n |
\n| Concavidade do gr\u00e1fico de $f$<\/td>\n | $ \\cap $<\/td>\n | P.I.<\/td>\n | $ \\cup $<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n- Representa\u00e7\u00e3o gr\u00e1fica<\/strong><\/li>\n<\/ul>\n
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