Matemática A · year 12 · Cálculo diferencial

Limites, indeterminações, limites notáveis e regra de L’Hôpital

66 exercises from national exams, newest first.

Exercise 2.1

1st phase 2025Open answer

Show the shared statement

Source: IAVE/EduQA — exam 635, 1st phase, 2025

Exercise 5.2

1st phase 2024Open answer

Show the shared statement

Exercise 5.2 — 5.2. Sabe-se que a função g é contínua em x = 1 . Determine o valor de k .

Source: IAVE/EduQA — exam 635, 1st phase, 2024

Exercise 7

1st phase 2024Open answer

$Exercise 7 — 7. Resolva este item sem recorrer à calculadora. Considere a função f , de domínio \ 0 R " , , definida por cos f x x x 1 4 ] g Estude a função f quanto à existência de assíntotas verticais ao seu gráfico e, caso…$

Source: IAVE/EduQA — exam 635, 1st phase, 2024

Exercise 6.2

2nd phase 2024Open answer

Show the shared statement

Source: IAVE/EduQA — exam 635, 2nd phase, 2024

Exercise 3.1

Special sitting 2024Open answer

Show the shared statement

Source: IAVE/EduQA — exam 635, Special sitting, 2024

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Exercise 4.1

1st phase 2023Open answer

Show the shared statement

Source: IAVE/EduQA — exam 635, 1st phase, 2023

Exercise 7

2nd phase 2023Multiple choice

Source: IAVE/EduQA — exam 635, 2nd phase, 2023

Exercise 7.1

Special sitting 2023Open answer

Show the shared statement

Exercise 7.1 — 7.1. Determine o valor de a para o qual existe lim f x x 0 " ] g .

Source: IAVE/EduQA — exam 635, Special sitting, 2023

Exercise 8.1

1st phase 2022Open answer

Show the shared statement

Source: IAVE/EduQA — exam 635, 1st phase, 2022

Exercise 8

2nd phase 2022Open answer

$Exercise 8 — 8. Resolva este item sem recorrer à calculadora. Seja f a função, de domínio R , definida por se se cos ln f x x x x e x x 1 0 0 1 $ ^ h Z [ \ ]] ]] Averigue se a função f é contínua em x 0 = .$

Source: IAVE/EduQA — exam 635, 2nd phase, 2022

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Exercise 10

Special sitting 2022Open answer

$Exercise 10 — 10. Seja f a função, de domínio R , definida por se se f x e x x x 1 3 0 5 3 0 x 5 ! ] g Z [ \ ]] ] Averigue, sem recorrer à calculadora, se a função f é contínua em x 0 = .$

Source: IAVE/EduQA — exam 635, Special sitting, 2022

Exercise 10.1

1st phase 2021Open answer

Show the shared statement

Source: IAVE/EduQA — exam 635, 1st phase, 2021

Exercise 5

2nd phase 2021Multiple choice

$Exercise 5 — 5. Na Figura 2, está representada parte do gráfico de uma função g, de domínio \ 2 R " , A reta de equação x 2 = é uma assíntota vertical ao gráfico da função g Seja vn ^ h a sucessão de termo geral v n 2 3 5 n A que é…$

Source: IAVE/EduQA — exam 635, 2nd phase, 2021

Exercise 12

2nd phase 2021Open answer

$Exercise 12 — 12. Resolva este item sem recorrer à calculadora. Considere, para um certo número real k, a função g, de domínio \ 0 R " , , definida por se se ln g x x x x x k x x x x 0 2 0 2 3 1 2 ^ h Z [ \ ]] ]] Sabe-se que existe…$

Source: IAVE/EduQA — exam 635, 2nd phase, 2021

Exercise 6

Special sitting 2021Multiple choice

Source: IAVE/EduQA — exam 635, Special sitting, 2021

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Exercise 10.1

Special sitting 2021Open answer

Show the shared statement

Source: IAVE/EduQA — exam 635, Special sitting, 2021

Exercise 12.1

Special sitting 2021Multiple choice

Show the shared statement

Source: IAVE/EduQA — exam 635, Special sitting, 2021

Exercise 1.2

1st phase 2020Multiple choice

Show the shared statement

Exercise 1.2 — 1.2. Seja f a função, de domínio , 8 3 − 6 @ , definida por log f x x 8 2 ] ] g g A que é igual lim f un ^ h ? (A) 3 − (B) 0 (C) 1 (D) 3 +

Source: IAVE/EduQA — exam 635, 1st phase, 2020

Exercise 10.1

1st phase 2020Open answer

Show the shared statement

Source: IAVE/EduQA — exam 635, 1st phase, 2020

Exercise 11.1

2nd phase 2020Open answer

Show the shared statement

Source: IAVE/EduQA — exam 635, 2nd phase, 2020

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Exercise 10.1

Special sitting 2020Multiple choice

Show the shared statement

Exercise 10.1 — 10.1. Sabe-se que g é contínua no ponto 1 Qual é o valor de k ? (A) 6 1 (B) 7 1 (C) 8 1 (D) 9 1

Source: IAVE/EduQA — exam 635, Special sitting, 2020

Exercise 13.2

1st phase 2019Open answer

Show the shared statement

Exercise 13.2 — 13.2. Averigue se a função f é contínua no ponto 0 Justifique a sua resposta.

Source: IAVE/EduQA — exam 635, 1st phase, 2019

Exercise 11.2

2nd phase 2019Multiple choice

Show the shared statement

Exercise 11.2 — 11.2. Qual é, para qualquer número real positivo a, o limite da sucessão ln n n a n 2 + + c m ? (A) a2 (B) a 2 (C) a (D) a

Source: IAVE/EduQA — exam 635, 2nd phase, 2019

Exercise 8

Special sitting 2019Multiple choice

Source: IAVE/EduQA — exam 635, Special sitting, 2019

Exercise 14.1

Special sitting 2019Open answer

Show the shared statement

Exercise 14.1 — 14.1. Determine lim x f x r − x 0 " ^ h

Source: IAVE/EduQA — exam 635, Special sitting, 2019

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Exercise 12.2

1st phase 2018Open answer

Show the shared statement

Exercise 12.2 — 12.2. Averigue se a função g é contínua no ponto 0 Justifique a sua resposta.

Source: IAVE/EduQA — exam 635, 1st phase, 2018

Exercise 14.1

Special sitting 2018Open answer

Show the shared statement

Source: IAVE/EduQA — exam 635, Special sitting, 2018

Exercise Part I · 4

1st phase 2017Multiple choice

Source: IAVE/EduQA — exam 635, 1st phase, 2017

Exercise Part II · 5.1

1st phase 2017Open answer

Show the shared statement

Source: IAVE/EduQA — exam 635, 1st phase, 2017

Exercise Part I · 3

2nd phase 2017Multiple choice

Source: IAVE/EduQA — exam 635, 2nd phase, 2017

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Exercise Part II · 4.1

Special sitting 2017Open answer

Show the shared statement

Exercise Part II · 4.1 — 4.1. Indique, justificando, se a seguinte afirmação é verdadeira ou é falsa. «A função f é contínua à esquerda no ponto 1, mas não é contínua à direita nesse ponto.»

Source: IAVE/EduQA — exam 635, Special sitting, 2017

Exercise Part I · 3

1st phase 2016Multiple choice

Source: IAVE/EduQA — exam 635, 1st phase, 2016

Exercise Part I · 4

1st phase 2016Multiple choice

Source: IAVE/EduQA — exam 635, 1st phase, 2016

Exercise Part I · 4

2nd phase 2016Multiple choice

Source: IAVE/EduQA — exam 635, 2nd phase, 2016

Exercise Part II · 5.2

2nd phase 2016Open answer

Show the shared statement

Exercise Part II · 5.2 — 5.2. Determine lim e x 1 600 x nx 0 − " − , em função de n, e interprete o resultado no contexto da situação descrita.

Source: IAVE/EduQA — exam 635, 2nd phase, 2016

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Exercise Part I · 6

Special sitting 2016Multiple choice

$Exercise Part I · 6 — 6. Para um certo número real k, é contínua em R a função f definida por se sen f x x x x k x 4 4 3 3 1 2 1 = + + = − + = − se Y ^ ^ h h Z [ \ ]] ] Qual é o valor de k ? (A) 3 5 − (B) 4 5 − (C) 4 5 (D) 3 5$

Source: IAVE/EduQA — exam 635, Special sitting, 2016

Exercise Part I · 8

Special sitting 2016Multiple choice

Source: IAVE/EduQA — exam 635, Special sitting, 2016

Exercise Part II · 4.1

Special sitting 2016Open answer

Show the shared statement

Exercise Part II · 4.1 — 4.1. Determine lim f x x x − " 3 + ^ h 8 B Interprete o valor obtido em termos de assíntotas do gráfico de f

Source: IAVE/EduQA — exam 635, Special sitting, 2016

Exercise Part I · 4

1st phase 2015Multiple choice

Source: IAVE/EduQA — exam 635, 1st phase, 2015

Exercise Part I · 4

2nd phase 2015Multiple choice

$Exercise Part I · 4 — 4. Para um certo número real k, é contínua em R a função f definida por ln f x e x x x x x 2 0 2 1 0 se se x k 2 # = + + + + ^ ] h g Z [ \ ]] ]] Qual é o valor de k ? (A) 0 (B) 1 (C) ln2 (D) ln 3$

Source: IAVE/EduQA — exam 635, 2nd phase, 2015

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Exercise Part I · 5

Special sitting 2015Multiple choice

Source: IAVE/EduQA — exam 635, Special sitting, 2015

Exercise Part I · 3

1st phase 2014Multiple choice

Source: IAVE/EduQA — exam 635, 1st phase, 2014

Exercise Part II · 5.1

1st phase 2014Open answer

Show the shared statement

Source: IAVE/EduQA — exam 635, 1st phase, 2014

Exercise Part I · 4

2nd phase 2014Multiple choice

Source: IAVE/EduQA — exam 635, 2nd phase, 2014

Exercise Part I · 5

2nd phase 2014Multiple choice

$Exercise Part I · 5 — 5. Considere, para um certo número real k, a função f, contínua em , 4 2 r r ; E, definida por cos f x x x x k x 2 4 2 3 2 se se 1 # r r r r = − − = ^ h Z [ \ ] ]] ] ] Qual é o valor de k ? (A) 0 (B) 1 (C) 2 (D) 4$

Source: IAVE/EduQA — exam 635, 2nd phase, 2014

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Exercise Part I · 3

Special sitting 2014Multiple choice

Source: IAVE/EduQA — exam 635, Special sitting, 2014

Exercise Part I · 6

Special sitting 2014Multiple choice

Source: IAVE/EduQA — exam 635, Special sitting, 2014

Exercise Part II · 5.1

Special sitting 2014Open answer

Show the shared statement

Source: IAVE/EduQA — exam 635, Special sitting, 2014

Exercise Part I · 4

1st phase 2013Multiple choice

Show the shared statement

$Exercise Part I · 4 — 4. Seja f a função, de domínio 0 \ R " ,, definida por sen f x x x = − ^ ^ h h Considere a sucessão de números reais xn ^ h tal que x n 1 n = Qual é o valor de lim f xn ^ h? (A) -1 (B) 0 (C) 1 (D) 3 +$

Source: IAVE/EduQA — exam 635, 1st phase, 2013

Exercise Part I · 5

1st phase 2013Multiple choice

Show the shared statement

Exercise Part I · 5 — 5. Seja f uma função de domínio R+ Sabe-se que lim ln x x f x 3 1 x + = " 3 + ^ h Qual das equações seguintes pode definir uma assíntota do gráfico da função f ? (A) y x 3 1 = (B) y x 3 2 = (C) y x = (D) y x 3 =

Source: IAVE/EduQA — exam 635, 1st phase, 2013

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Exercise Part II · 4.1

2nd phase 2013Open answer

Show the shared statement

Source: IAVE/EduQA — exam 635, 2nd phase, 2013

Exercise Part I · 5

Special sitting 2013Multiple choice

Show the shared statement

Exercise Part I · 5 — 5. Seja f uma função de domínio R Sabe-se que: • lim ( ) f x 1 x = " 3 + • ( ) lim f x x 2 2 x + = " 3 - 6 @ Em qual das opções seguintes pode estar representada parte do gráfico da função f ? (A) (B) (C) (D) y x O 2 1…

Source: IAVE/EduQA — exam 635, Special sitting, 2013

Exercise Part II · 5.1

Special sitting 2013Open answer

Show the shared statement

Exercise Part II · 5.1 — 5.1. Determine k de modo que lim f x f 0 x 0 = " - ^ ^ h h

Source: IAVE/EduQA — exam 635, Special sitting, 2013

Exercise Part II · 6

Special sitting 2013Open answer

Show the shared statement

Exercise Part II · 6 — 6. Considere duas funções g e h, de domínio R + Sabe-se que: • a reta de equação y = 2x - 1 é assíntota do gráfico da função g • a função h é definida por h x x g x 1 2 = − 2 ^ ` h j ; E Mostre que o gráfico da função h…

Source: IAVE/EduQA — exam 635, Special sitting, 2013

Exercise Part I · 4

2nd phase 2012Multiple choice

Show the shared statement

Exercise Part I · 4 — 4. Na Figura 1, está representada, num referencial o.n. xOy, parte do gráfico de uma função f , de domínio ,1 3 - 6 @ Sabe-se que: • f 1 4 = − ^ h • a reta de equação x 1 = é assíntota do gráfico de f • xn ^ h é uma…

Source: IAVE/EduQA — exam 635, 2nd phase, 2012

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Exercise Part II · 4.1

2nd phase 2012Open answer

Show the shared statement

Exercise Part II · 4.1 — 4.1. Determine k, de modo que lim f x f 0 x 0 = " + ^ ^ h h

Source: IAVE/EduQA — exam 635, 2nd phase, 2012

Exercise Part I · 5

Special sitting 2012Multiple choice

Show the shared statement

Exercise Part I · 5 — 5. Sejam f e g funções de domínio , 0 3 + 6 @ Sabe-se que: • a reta de equação y 3 = é assíntota horizontal do gráfico de f • f não tem zeros; • g x f x e 3 x = − - ^ ^ h h Qual das opções seguintes define uma assíntota…

Source: IAVE/EduQA — exam 635, Special sitting, 2012

Exercise Part II · 5.1

Special sitting 2012Open answer

Show the shared statement

Source: IAVE/EduQA — exam 635, Special sitting, 2012

Exercise Part I · 3

1st phase 2011Multiple choice

Show the shared statement

Exercise Part I · 3 — 3. Na Figura 1, está representada, num referencial o. n. xOy , parte do gráfico de uma função g , de domínio ,3 3 − + 7 A g O x y Figura 1 A recta de equação y = 2x - 4 é assimptota do gráfico de g Qual das afirmações…

Source: IAVE/EduQA — exam 635, 1st phase, 2011

Exercise Part I · 5

1st phase 2011Multiple choice

Show the shared statement

Exercise Part I · 5 — 5. Qual é o valor de lim x x x →             0 2 2 1 2 sen ? (A) 4 (B) 0 (C) 4 1 (D) 2 1

Source: IAVE/EduQA — exam 635, 1st phase, 2011

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Exercise Part I · 5

2nd phase 2011Multiple choice

Show the shared statement

Exercise Part I · 5 — 5. Para um certo número real positivo, k, a função g definida em  por g x x x x k x x ( ) sen ln( ) = > − ≤       3 0 0 se se é contínua. Qual é o valor de k ? (A) e 3 (B) e3 (C) e—3 (D) 3e

Source: IAVE/EduQA — exam 635, 2nd phase, 2011

Exercise Part II · 5.1

Special sitting 2011Open answer

Show the shared statement

Source: IAVE/EduQA — exam 635, Special sitting, 2011

Exercise Part II · 6.1

Special sitting 2011Open answer

Show the shared statement

Exercise Part II · 6.1 — 6.1. Calcule o valor de lim sen f x x x 0 p - " _ i

Source: IAVE/EduQA — exam 635, Special sitting, 2011

Exercise Part I · 5

1st phase 2010Multiple choice

Show the shared statement

Exercise Part I · 5 — 5. Na Figura 2, está representada, num referencial o.n. xOy , parte do gráfico de uma função f , contínua, de domínio Tal como a Figura 2 sugere, a recta de equação x = 1 é assimptota do gráfico de f Figura 2 Qual é o…

Source: IAVE/EduQA — exam 635, 1st phase, 2010

Exercise Part II · 6.2

1st phase 2010Open answer

Show the shared statement

Source: IAVE/EduQA — exam 635, 1st phase, 2010

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Exercise Part I · 5

2nd phase 2010Multiple choice

Show the shared statement

Exercise Part I · 5 — 5. Considere a função g , de domínio R, definida por Considere a sucessão de termo geral Qual é o valor de ? (A) (B) 1 (C) 0 (D) –∞ +∞ lim ( ) n n g u →+∞ 1 n u n = se ( ) ln se x e x g x x x  ≤0  =  > 0 

Source: IAVE/EduQA — exam 635, 2nd phase, 2010

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