Scarica Esercizi di Calcolo: Limiti di Funzioni - Politecnico di Torino - Prof. Spreafico e più Esercizi in PDF di Calcolo I solo su Docsity!
II Faculty of Engeneering- Politecnico di Torino - 2007/
Exercise session for Calculus I (Professor: Maria Luisa Spreafico)
SESSION 4- LIMITS OF FUNCTIONS
- Find the indicated limit or explain why it does not exist.
a) limx→+∞ x √^4 −x^3 + x+x^2 −x^3 b)^ limx→−∞^
x^3 3 x^2 − 4 −^
x^2 3 x+2 c)^ limx→^0
x^3 −x^2 +4x x^5 −x d) limx→ 1 + x (^4) −x (^3) + 1 −x^3 e)^ limx→+∞
x(
x + 1 −
x) f ) limx→ 0
√ (^3) 1+x− √ (^31) −x x g) limx→π sinx−^ xπ h) limx→+∞ x√+cosx− 1 x i) limx→+∞ x x++M√^ (xx)
- Find the indicated limit or explain why it does not exist.
a) limx→+∞ 2
2 x+2−x (2x−1)^2 a)^ limx→−∞^
22 x+2−x (2x−1)^2 c)^ limx→−∞^ √^2 x−^1 3 x^2 − 2 d) limx→ 2 2 x
(^3) − 5 x (^2) − 4 x+ x^4 − 4 x^3 +5x^2 − 4 x+4 e)^ limx→^0 arctan^
1 x f^ )^ limx→^0 √sin 5x^2 1+x^2 − 1 g) limx→− 1 +^ E(x^3 + 1) g) limx→− 1 −^ E(x^3 + 1) i) limx→ 1 x−^
1 log x l) limx→+∞ √^23 xx− (^2) −^12 m) limx→+∞
√5+cos x x^2 +1 n)^ limx→^0 tan(x^ +^
π 2 )
- Find the indicated limit or explain why it does not exist.
a) limx→+∞ log(1+xe
x) e−^3 x− 1 b)^ limx→^0
log(1+xex) e−^3 x− 1 c)^ limx→^0
√1+x (^2) − 1 3 x^2 d) limx→+∞ log(1+xe
x) e−^3 x^ e)^ limx→^0
1 −log(e+x) x f^ )^ limx→^0
√ √4+3x−^2 9+2x− 3 g) limx→ 0 + (xx^ + x−^ (^1) x ) h) limx→ (^3) xsin(3− 3 e−x−x) 3 i) limx→+∞ x
(^2) log (^3) x+x log (^7) x 1+x^3
- Evaluate the limits as x approaches 0, for:
a) sin^ x
4 sin^2 x^2 b)^
√ 1 −x (^5) −√1+x 5 sin^5 x c)^
x+sin 4x x+sin x d)^
sin(sin x) x e) (^) xx−+sin2 sin^ x x f ) (^) x^15 −−cos 2sin 3 x (^) x g) e
3 x− 1 2 x^2 h)^
1 −√cos x 2 x^2 +x^3
- Evaluate the limits as x approaches +∞ for:
a) (cos (^1) x )x
2 b) e−x(e + (^2) x )x^ c)
1 − cos (^) x^52
(x^4 + arctan 3x)
- Find the indicated limits (if there exist):
a) limx→+∞(
x + cos x) b) limx→+∞
x cos x c) limx→+∞
x sin (^1) x d) limx→+∞ xM (x)
- Determine λ ∈ R such that
lim x→−∞
x^2 − 1(
x^2 + λ + x) = 2 1
- Find (if there exist) the values of α such that the following functions are con- tinuous.
a) f 1 (x) =
{ (^) x (^2) −1+α 2 x^2 +1 se^ x^ ≤^0 x^2 + α se x > 0 ,
b) f 2 (x) =
sin(x + α) se x < 0 cos(x − α) se x ≥ 0
c) f 3 (x) =
sin αx se x > 0 x cos αx se x ≤ 0. d) f 4 (x) =
sin(x^2 ) x(√1+x−1) se^ x >^0 E(x) + α se x < 0 α − 1 se x = 0.
- Determine which of the following functions are continuous on R:
a) f 1 (x) =
arctan (^1) x
∣ (^) se x 6 = 0 π/ 2 se x = 0 , b) f 2 (x) =
x^2 + 3x − 1 se x < 0 − cos x se x ≥ 0
c) f 3 (x) =
4 x − (^) x^1 se x < 0 e x+1^1 se x > 0 , d) f 4 (x) =
sin (^) x^1 se x < (^2) π 1 se x ≥ (^2) π.
- Find the infinitesimal order α and the principal part kxα^ with respect to x for x → 0 , of the following functions:
a) f (x) = e^3 x
4 − 1 b) f (x) =
1 + x −
1 − x c) f (x) = (1 + 3x^2 )^3 − 1 d) f (x) = x^2 + sin(2x^3 ) e) f (x) = log(x + 3) − log 3 f ) f (x) = log(cos x)
- Find the infinitesimal order α and the principal part (^) xkα with respect to (^) x^1 for x → +∞, for the following functions:
a) f (x) = 2 x
(^2) + √ (^3) x x^3 b)^ f^ (x) = log^
x+ x+1 c)^ f^ (x) =^ e^
x+ x (^) − e
- Find the infinitesimal order α and the principal part k(x − x 0 )α^ with respect to (x − x 0 ) for x → x 0 , for the following functions:
a) f (x) = log x − log 2 (x 0 = 2) b) f (X) = ex^ − e (x 0 = 1) c) f (x) = 1 − sin x (x 0 = π 2 )
- Identify any horizontal, vertical and oblique asymptotes of the considered functions.
a) f (x) =
x^2 − 1 b) f (x) = log(x^2 − 1) c) f (x) = log(1 + e^2 x) d) f (x) = xex+1^ e) f (x) = |x^2 − 1 | + x^2 − x f ) f (x) = (^) xx 2 +5− 4 g) f (x) = log(ex^ − x) h) f (x) = 2 − 2 e−|x|^ − x i) f (x) = |x|e
1+x 2+x
a) write the definition of limited function; b) give an exmple of non limited function in the interval [56, 57]; c) give an example of limited and increasing function; d) show that if f (x) is a limited function, then g(x) = ef^ (x)^ is limited, too; e) give an example of a limited function f (x) such that g(x) = log |f (x)| is non limited.
f) f (x) ≤ 0 ∀x ∈ Domf (x)
- Give an example of a function f (x) such that: a) x = 1 and x = 5 are vertical asymptots for f (x) b) y = 27 is the orizontal asymptote of f (x)
c) limx→ 1 f x^ (−x 1 ) = 0
d) limx→+∞ f^ e(xx )= 0 e) the domain of f (x) is limited f) the range of f (x) is limited g) the line y = − 3 x + 7 is the oblique asynptote of f (x) h) f (x) is increasing and limited 23. Give an example of a function f (x) such that:
a) f (x) is non continuous on the interval [− 6 , 0] but has an absolute maximum value in this interval; b) f (x) is continuous on the interval (7, 9) but does not admit absolute minimum value in this interval; e) f (x) is continuous on the interval [6, 10] and has exactly two minimal points in this interval.
Risultati
a) + ∞ b) 29 c) − 4 d) − ∞ e) 12 f ) (^23) g) − 1 h) + ∞ i) 1
a) 1 b) + ∞ c) − √^23 d) 75 e) non esiste f ) 10 g) 0 h) − 1 i) (^1) e l) √^23 m) 0 n) thelimitdoesnotexist
a) − ∞ b) − 13 c) (^16) d) + ∞ e) − (^1) e f ) (^94) g) + ∞ h) 12 i) 0
a) 1 b) − 1 c) 52 d) 1 e) − 2 f ) the limit does not exist g) the limit does not exist h) (^18)
a) e−^
(^12) b) e
(^2) e c) (^252)
a) + ∞ b) the limit does not exist c) 0 d) the limit does not exist
λ = 4
a) α = 1 ±
√ 5 2 b)^ α^ =^
π 4 +^ kπ k^ ∈^ Z^ c)^ α^ = 0^ d)^ α^ = 3
a) yes b) yes c) no d) yes
a) f (x) ∼ 3 x^4 b) f (x) ∼ x c) f (x) ∼ 9 x^2 d) f (x) ∼ x^2 e) f (x) ∼ x 3 f ) f (x) ∼ − x
2 2
a) f (x) ∼ (^2) x b) f (x) ∼ (^2) x c) f (x) ∼ ex
a) f (x) ∼ x− 2 2 b) f (x) ∼ e(x − 1) c) f (x) ∼ 12 (x − π 2 )^2
a) f (x) = E(x) b) f (x) = log(x − 7) c) f (x) = | |x − 8 | − 1 |
