(2)Determine se a afirmação é falsa ou verdadeira. Se for verdadeira, explique por quê. Caso contrário, explique por que ou dê um exemplo que mostre que é falsa. - (a) Se $\lim _{x \rightarrow 0} f(x)=\infty$ e $\lim _{x \rightarrow 0} g(x)=\infty$, então $\lim _{x \rightarrow 0}[f(x)-g(x)]=0$.
Passo 1
Vamos analisar a afirmação: se
e
, então
. Esse exercício pertence à matéria de Cálculo, mais especificamente ao estudo de limites de funções. Para resolver, vamos precisar entender como os limites de duas funções que tendem ao infinito se comportam quando subtraídos.
Passo 2
Primeiro, sabemos que
significa que, conforme
se aproxima de 0,
cresce sem limite, ou seja, fica cada vez maior. O mesmo vale para
.
Passo 3
A afirmação sugere que, se ambas as funções crescem para o infinito, a diferença entre elas vai tender a zero. Para verificar isso, precisamos pensar em como essas duas funções crescem em relação uma à outra.
Passo 4
Vamos considerar um exemplo para testar a afirmação. Seja
e
. Quando
se aproxima de 0, tanto
quanto
tendem a
. A diferença entre elas é:
Passo 5
Observamos que
tende a
quando
, não a 0. Isso mostra que, mesmo que ambas as funções
e
tendam a
, a diferença entre elas pode não tender a 0.
Resposta
A afirmação é falsa. Por exemplo, se
e
, então
.
Exercícios relacionados
<html><head></head><body><div>(4) (3)<br>
(a) Esboce o gráfico da função <span class="math-inline ">
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(a) Esboce o gráfico da função <span class="math-inline ">
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142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z”></path></g><g data-mml-node=“mo” transform=“translate(1438,0)”><path data-c=“29” d=“M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z”></path></g><g data-mml-node=“mo” transform=“translate(2104.8,0)”><path data-c=“3D” d=“M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z”></path></g><g data-mml-node=“msup” transform=“translate(3160.6,0)”><g data-mml-node=“mi”><path data-c=“1D465” d=“M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 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(2)Ache cada limite.
(2)Determine se a afirmação é falsa ou verdadeira. Se for verdadeira, explique por quê. Caso contrário, explique por que ou dê um exemplo que mostre que é falsa.
-© Se e
não existem, então
não existe.
-© Se
<html><head></head><body><div>(4) (3)<br>
-© Encontre uma fórmula para <span class="math-inline " data-overflow=“visible”>
<mjx-container class=“MathJax” jax=“SVG”><svg style=“vertical-align: -0.464ex;” xmlns=“http://www.w3.org/2000/svg” width=“1.707ex” height=“2.181ex” role=“img” focusable=“false” viewBox=“0 -759 754.5 964”><title>g’</title><g stroke=“currentColor” fill=“currentColor” stroke-width=“0” transform=“scale(1,-1)”><g data-mml-node=“math”><g data-mml-node=“msup”><g data-mml-node=“mi”><path data-c=“1D454” d=“M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z”></path></g><g data-mml-node=“TeXAtom” transform=“translate(510,363) scale(0.707)” data-mjx-texclass=“ORD”><g data-mml-node=“mi”><path data-c=“2032” d=“M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z”></path></g></g></g></g></g></svg></mjx-container></span>.</div>
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-© Encontre uma fórmula para <span class="math-inline " data-overflow=“visible”>
<mjx-container class=“MathJax” jax=“SVG”><svg style=“vertical-align: -0.464ex;” xmlns=“http://www.w3.org/2000/svg” width=“1.707ex” height=“2.181ex” role=“img” focusable=“false” viewBox=“0 -759 754.5 964”><title>g’</title><g stroke=“currentColor” fill=“currentColor” stroke-width=“0” transform=“scale(1,-1)”><g data-mml-node=“math”><g data-mml-node=“msup”><g data-mml-node=“mi”><path data-c=“1D454” d=“M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z”></path></g><g data-mml-node=“TeXAtom” transform=“translate(510,363) scale(0.707)” data-mjx-texclass=“ORD”><g data-mml-node=“mi”><path data-c=“2032” d=“M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z”></path></g></g></g></g></g></svg></mjx-container></span>.</div>
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