Calcule o limite, se existir. $\lim _{x \rightarrow-2} \frac{x^{2}-x-6}{3 x^{2}+5 x-2}$
Passo 1
Então, a gente tem que calcular o limite quando
se aproxima de
da função
. Vamos usar o conteúdo de limites de funções racionais, onde a gente substitui o valor de
na função, mas primeiro vamos ver se conseguimos simplificar essa fração.
Passo 2
Primeiro, vamos fatorar o numerador
. Procura dois números que multiplicados dão
e somados dão
. Esses números são
e
. Então, o numerador fica:
Passo 3
Agora, vamos fatorar o denominador
. Precisamos de dois números que multiplicados dão
e somados dão
. Esses números são
e
. Então, reescrevemos o denominador:
Passo 4
Com as fatorações, nossa expressão fica:
A gente pode cancelar o
que está no numerador e no denominador, já que
não está se aproximando de
de uma forma que torne o denominador zero.
Passo 5
Agora, a expressão simplificada é:
Agora, podemos substituir diretamente
na expressão simplificada.
Passo 6
Substituindo:
Resposta
O limite é
.
Exercícios relacionados
Calcule o limite, se existir.
Encontre, quando existir, o limite. Caso não exista, explique.
Seja
Encontre o valor de
tal que exista o
.
<html><head></head><body><div>Determine o valor de <span class="math-inline " data-overflow=“visible”>
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<mjx-container class=“MathJax” jax=“SVG”><svg style=“vertical-align: -0.477ex;” xmlns=“http://www.w3.org/2000/svg” width=“1.396ex” height=“1.498ex” role=“img” focusable=“false” viewBox=“0 -451 617 662”><title>gamma</title><g stroke=“currentColor” fill=“currentColor” stroke-width=“0” transform=“scale(1,-1)”><g data-mml-node=“math”><g data-mml-node=“mi”><path data-c=“1D738” d=“M5 269Q5 285 19 312T57 368T124 421T215 451H241Q274 451 303 439T353 406T389 361T416 311T432 266T442 232L444 220L446 216L450 226Q473 278 513 357T561 441Q566 444 584 444H594Q617 444 617 430Q617 426 596 389T536 273T462 110L452 84L451 70Q447 12 427 -76T388 -192Q375 -211 355 -211Q339 -211 332 -198T325 -171Q325 -114 386 64L393 84V98Q393 181 371 241Q360 280 319 303T210 327Q158 327 126 317T84 296T68 272T59 258Q55 256 36 256Q23 256 18 256T9 260T5 269Z”></path></g></g></g></svg></mjx-container></span> dado: a) <span class="math-inline ">
<mjx-container class=“MathJax” jax=“SVG”><svg style=“vertical-align: -0.489ex;” xmlns=“http://www.w3.org/2000/svg” width=“5.377ex” height=“1.995ex” role=“img” focusable=“false” viewBox=“0 -666 2376.6 882”><title>γ = 5</title><g stroke=“currentColor” fill=“currentColor” stroke-width=“0” transform=“scale(1,-1)”><g data-mml-node=“math”><g data-mml-node=“mi”><path data-c=“1D6FE” d=“M31 249Q11 249 11 258Q11 275 26 304T66 365T129 418T206 441Q233 441 239 440Q287 429 318 386T371 255Q385 195 385 170Q385 166 386 166L398 193Q418 244 443 300T486 391T508 430Q510 431 524 431H537Q543 425 543 422Q543 418 522 378T463 251T391 71Q385 55 378 6T357 -100Q341 -165 330 -190T303 -216Q286 -216 286 -188Q286 -138 340 32L346 51L347 69Q348 79 348 100Q348 257 291 317Q251 355 196 355Q148 355 108 329T51 260Q49 251 47 251Q45 249 31 249Z”></path></g><g data-mml-node=“mo” transform=“translate(820.8,0)”><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=“mn” transform=“translate(1876.6,0)”><path data-c=“35” d=“M164 157Q164 133 148 117T109 101H102Q148 22 224 22Q294 22 326 82Q345 115 345 210Q345 313 318 349Q292 382 260 382H254Q176 382 136 314Q132 307 129 306T114 304Q97 304 95 310Q93 314 93 485V614Q93 664 98 664Q100 666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 662Q397 666 402 666Q410 666 410 648V635Q328 538 205 538Q174 538 149 544L139 546V374Q158 388 169 396T205 412T256 420Q337 420 393 355T449 201Q449 109 385 44T229 -22Q148 -22 99 32T50 154Q50 178 61 192T84 210T107 214Q132 214 148 197T164 157Z”></path></g></g></g></svg></mjx-container></span></div>
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<mjx-container class=“MathJax” jax=“SVG”><svg style=“vertical-align: -0.477ex;” xmlns=“http://www.w3.org/2000/svg” width=“1.396ex” height=“1.498ex” role=“img” focusable=“false” viewBox=“0 -451 617 662”><title>gamma</title><g stroke=“currentColor” fill=“currentColor” stroke-width=“0” transform=“scale(1,-1)”><g data-mml-node=“math”><g data-mml-node=“mi”><path data-c=“1D738” d=“M5 269Q5 285 19 312T57 368T124 421T215 451H241Q274 451 303 439T353 406T389 361T416 311T432 266T442 232L444 220L446 216L450 226Q473 278 513 357T561 441Q566 444 584 444H594Q617 444 617 430Q617 426 596 389T536 273T462 110L452 84L451 70Q447 12 427 -76T388 -192Q375 -211 355 -211Q339 -211 332 -198T325 -171Q325 -114 386 64L393 84V98Q393 181 371 241Q360 280 319 303T210 327Q158 327 126 317T84 296T68 272T59 258Q55 256 36 256Q23 256 18 256T9 260T5 269Z”></path></g></g></g></svg></mjx-container></span> para cada <span class="math-inline " data-overflow=“visible”>
<mjx-container class=“MathJax” jax=“SVG”><svg style=“vertical-align: -0.477ex;” xmlns=“http://www.w3.org/2000/svg” width=“1.396ex” height=“1.498ex” role=“img” focusable=“false” viewBox=“0 -451 617 662”><title>gamma</title><g stroke=“currentColor” fill=“currentColor” stroke-width=“0” transform=“scale(1,-1)”><g data-mml-node=“math”><g data-mml-node=“mi”><path data-c=“1D738” d=“M5 269Q5 285 19 312T57 368T124 421T215 451H241Q274 451 303 439T353 406T389 361T416 311T432 266T442 232L444 220L446 216L450 226Q473 278 513 357T561 441Q566 444 584 444H594Q617 444 617 430Q617 426 596 389T536 273T462 110L452 84L451 70Q447 12 427 -76T388 -192Q375 -211 355 -211Q339 -211 332 -198T325 -171Q325 -114 386 64L393 84V98Q393 181 371 241Q360 280 319 303T210 327Q158 327 126 317T84 296T68 272T59 258Q55 256 36 256Q23 256 18 256T9 260T5 269Z”></path></g></g></g></svg></mjx-container></span> dado: a) <span class="math-inline ">
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