Tutorial 4

Instructions: No Calculators; Show all workings

1. Evaluate the following limits at infinity if they exist.If any of them fail to exist, say so and say
why.
√
3x6 − 7x5 + x 5x + 2x3 4x2 + 3x − 7
(a) lim (b) lim (c) lim
x→−∞ 5x6 + 4x5 − 3 x→−∞ x3 + x − 7 x→−∞ 7 − 3x
x2 1 + 3x p p 
(d) lim 4x (e) lim √ (f) lim x2 − x + 1 − x2 + 1
x→∞ e − 1 − 4x x→−∞ 2x2 + x x→−∞

ln x u √ √ √ 
(g) lim √ (h) lim √ (i) lim x x + 3 − x − 2
x→∞ x u→∞ u2 + 1 x→∞
√
3x + |1 − 3x| √ √ x2 + 4x
 q q 
(j) lim (k) lim x+ x− x− x (l) lim
x→∞ 1 − 5x x→∞ x→−∞ 4x + 1
√
8 4
3x + 7x + 10 h p p i 3x
(m) lim (n) lim x2 + 5x − x2 − x (o) lim √
x→∞ x4 − 2x2 + 1 x→∞ x→∞ 2
4x + x − 2x
√ √3
√
p  4x + 2 x2 + x − 4 x4 + 5
(p) lim x2 + x − x (q) lim (r) lim
x→∞ x→∞ 3x + 4 x→∞ x+1
x+1 3x + 5  
(s) lim √ (t) lim √ (t) lim sin 2 tan−1 x)
x→∞ x2 x→∞ x2 + 5 − x x→∞

2. Evaluate the following limits if they exist. If any of them fail to exist, say so and say why.
√ √
x2 − 16 6 − 3x x+2− 2
(a) lim (b) lim− (c) lim
x→4 |x − 4| x→2 |x − 2| x→0 x
! 1
! 8x
1
1 1 x3 − 2
(d) lim − (e) lim (f) lim 1 + 3x
x→0 x x2 + x x→8 x − 8 x→0

ln(2 + 2x) − ln 2 1 − e−x x2 − 16

(g) lim− (h) lim (i) lim ln |x|
x→2 x x→0 1 − x x→−4 x + 4
√  x1
x x2 − x + 4 − 2 
(j) lim √ (k) lim (l) lim 1 − 2x
x→0 4−x−2 x→0 x2 + 3x x→0
r
2w2 − 50 3 2
x +x +x+1 1 1 
(m) lim (n) lim (o) lim −
x→5 (w − 5)(w − 1) x→−1 3x + 3 x→0− x |x|
√ 3
x2 + 9x + 9 − 3 (2 + h) − 8  
(p) lim (q) lim (r) lim 2x + |x − 3|
x→0 x h→0 h x→3
√
x+2 x2 + 12 − 4 x+2
(s) lim √ (t) lim (u) lim √
x→−2 x2 + 5 − 3 x→2 x−2 x→−2 2
x +5−3
(2 + h)3 − 8   1 1 
(q) lim (r) lim 2x + |x − 3| (s) lim −
x→0 h x→3 x→0− x |x|
3. Evaluate the following trigonometric limits if they exist. If any of them fail to exist, say so and
say why.
sin 3x 2x arcsin 3x
(a) lim (b) lim (b) lim
x→0 sin 5x x→0 tan x x→0 arcsin 5x !
x 1 − cos x 1 1
(d) lim (e) lim (f) lim −
x→0 x − sin 2x x→0 sin 2x x→0 x2 tan x
2x sin x sec x − cos x  sin(x − 1) 
(g) lim (h) lim (i) lim
x→0 1 − cos x x→0 x2 x→1 x2 + 2x − 3
2 sin 4x + 1 − cos x x2 − 9 2
4x − 2 sin x
(j) lim (k) lim (l) lim
x→0 x x→3 sin(x − 3) x→0 x
2x sin x tan 3x − sin 3x  x 
(m) lim (n) lim (o) lim sin−1
x→0 1 − cos x x→0 x3 x→+∞ 1 + 2x
tan x
(s) lim−
x→ π2
sec x

4. Consider the following piece-wise defined function:

(
2x2 − 3x + 1 if x < −2,
f (t) =
−x2 + 3x + 25 if x > −2,

Find the following limits:

(a) lim f (x) (b) lim f (x) (c) lim f (x) (d) lim f (x) (e) lim f (x)
x→−2− x→−2+ x→−2 x→−3 x→−1

5. Consider the following piece-wise defined function:

(
−2 + 3x if x < 2,
f (t) = 2
−2x − 2x + 19 if x > 2,

Find the following limits:

(a) lim f (x) (b) lim f (x) (c) lim f (x) (d) lim f (x) (e) lim f (x)
x→2− x→2+ x→2 x→3 x→1

6. Consider the following piece-wise defined function:


 x + 1 if
 x < 2,
f (t) = x2 − 2 if 2 < x < 4,
 √

x + 5 if x ≥ 4,

Find the following limits:

(a) lim f (x) (b) lim f (x) (c) lim f (x) (d) lim f (x) (e) lim f (x)
x→1+ x→2+ x→2− x→3+ x→−1

(a) lim f (x) (b) lim f (x) (c) lim f (x) (d) lim f (x) (e) lim f (x)
x→3− x→3 x→1− x→1 x→4+

(a) lim f (x) (b) lim f (x) (c) lim f (x) (d) lim f (x) (e) lim f (x)
x→4− x→4 x→2 x→0 x→11

7. If
f (x)
lim =4
x→0 x2
Find limx→0 f (x).

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 8. Given that lim f (x) = 7 and lim g(x) = −6 find the limits that exist. If the limit does not
x→−∞ x→−∞
exist, explain why.
h i
(a) lim 2f (x) − g(x)
x→−∞
h i
(b) lim x2 g(x)
x→−∞

g(x) i
(c) lim [f (x) +
x→−∞ x
h xf (x) i
(d) lim
x→−∞ (2x + 3)g(x)

xf (x) + 3
9. Suppose lim f (x) = −1. Evaluate lim
x→1 x→1 2f (x) + 1
10. Find the value of the constant a for which
x2 + ax + 3
lim
x→−2 x2 + x − 2

exists and find the limit.

11. Is there a number b such that

3x2 + bx + b + 3
lim
x→−2 x2 + x − 2
exists? If so, find the value of b and the value of the limit.

12. Is there a number b such that

bx2 + 15x + 15 + b
lim
x→−2 x2 + x − 2
exists? If so, find the value of b and the value of the limit.

13. Find constants a and b such that √

ax + b − 2
lim=1
x x→0

 1 a 
14. Find all values of a such that lim − 2 exists and is finite.
x→1 x − 1 x −1
15. Let
2x2 − 3x
f (x) =
|2x − 3|
(a) Find lim
+ f (x) (b) lim
− f (x)
3 3
x→ 2 x→ 2

Does lim 3

f (x) exist? Provide a reason.
x→ 2

f (x) − 8
16. If lim = 10 find lim f (x)
x→1 x−1 x→1

f (x)
17. If lim = 5, find the following limits
x→0 x2
(a) lim f (x)
x→0
f (x)
(b) lim
x→0 x

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