Practice with Limits
Evaluate the following limits
(1) lim (6x2 − 4x + 3)
x4 − 81
x→3 x − 3
x→1
(8) lim
x2 − 49
x→7 x − 7
(9) lim ((x2 − 2)2 + 6)
(2) lim
x→0
x2 − 6x + 8
x→2
x−2
(10) lim
2x2 + 9x − 5
x→−5
x+5
(11) lim
x3 − 1
x→1 x − 1
(12) lim
(3) lim
x→0
5x
x
17x
x→0 2x
(4) lim
(5) lim
x→0
−317x
422x
−317x − 3
x→0 422x + 5
x2 − 4x + 3
(6) lim 2
x→3 x − 2x − 3
(13) lim
x3 + 8
(7) lim
x→−2 x + 2
(14) lim
x→∞
1
x+2
x−2
2x3 − 5x + 7
x→∞ 7x3 + 2x2 − 6
3x2 + 2x − 5
x→∞ 5x2 + 3x + 1
(17) lim
x2 − 7x + 11
x→∞
3x2 + 10
(18) lim
(15) lim
(3x − 1)(4x − 5)
x→∞ (x + 6)(x − 3)
(16) lim
Show the following equalities are true:
√
√
3+x− 3
1
(1) lim
= √ (hint: multiply the top and the bottom by the conjugate of
x→0
x
2 3
√
√
√3 )
the numerator. So, multiply the expression by √3+x+
3+x+ 3
√
√
x+h− x
1
(2) lim
= √
h→0
h
2 x
x
1
=
(3) lim √
2
x→∞
2
4x + 1 − 1
(4) lim
x→∞
√
n2 + 1 − n = 0 (hint: multiply by the fraction
over the conjugate)
2
√
2
√n +1+n .
n2 +1+n
This is the conjugate
Consider the piecewise function
 1
if x < −1

x2





2
if − 1 ≤ x < 1


3
if x = 1
f (x) =



x+1
if 1 < x ≤ 2




 −1
if x > 2
(x−2)2
First, sketch the graph of this function, then determine the following limits.
3
2
1
-2
-1
1
-1
-2
(1)
lim f (x) =
x→−1−
(2) lim + f (x) =
x→−1
(3) lim f (x) =
x→−1
(4) lim− f (x) =
x→1
(5) lim+ f (x) =
x→1
(6) lim f (x) =
x→1
(7) lim− f (x) =
x→2
(8) lim+ f (x) =
x→2
(9) lim f (x) =
x→2
(10) lim f (x) =
x→−3
(11) lim f (x) =
x→5
(12) lim f (x) =
x→1.5
3
2
3