Basics of Calculus
Limits
First Session
2021-2022
Mr. Mohammed Mounir
Basics
Calculator
2 1
+ =
7 3
....
2+1
7+3
𝐵)
𝐴)
2×3+1×7
7×3
𝐶)
2×1
7×3
𝐷)
2+1
7×3
1
8
× 440
𝐴) 0
1
+
2
× 2000 − 4 × 825 =
𝐵)1000
𝐶)210
𝐷)4030
1
8
1
× 4401 +
𝐴) 2800
32
× 2804
𝐵) 4401
=
𝐶) 8200
𝐷) 𝑈𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑
Let
𝑓(𝑥) = 𝑥 + sin 𝑥 , the value of 𝑓(2) to the nearest 0.01 =
𝐴)2.03
𝐵)2.91
𝐶)2.50
𝐷) 𝑛𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒𝑠𝑒
Degree
Gradian
Radian
Undefined quantity
7
0
Unspecified quantities
0
0
,
∞
∞
,
0×∞,
∞−∞,
1∞ ,
00 ,
∞0
EX.
0
0
EX.
∞−∞
Extended real number
EX.
1
=
∞
EX.
Log 0 =
EX.
𝑒∞ =
EX. Sin ∞ =
EX.
Log ∞ =
EX.
𝑒 −∞ =
Function
EX.
𝑥2 − 1
𝑓𝑥 =
𝑥−1
EX.
𝑓 0 =
𝑓 4 =
𝑓 2 =
𝑓 2 =
𝑓 1 =
𝑓 0 =
The limits
The right limit and the left limit (graphically)
limit from the right
limit from the left
For a limit to exist, the limit from the left
and the limit from the right must exist and
be equal.
EX.
EX.
EX.
The right limit and the left limit (Numerically)
Explore lim
x→5
x −1 − 2
use the calculator
x−5
Finding the Limit of a Function (Algebraically)
Direct substitution
EX.
EX.
EX.
lim 𝑥 − 3 =
EX. 𝑥→3
𝐴) 0
𝐵) 1
𝐶)3
𝐷) 𝐷. 𝑁. 𝐸
𝑥−3
=
EX. 𝑥lim
→3 𝑥 − 3
𝐴) 0
𝐵) 1
𝐶) − 1
𝐷) 𝐷. 𝑁. 𝐸
EX.
lim
𝑥→ 0
1
1
1 + 𝑒𝑥
𝐴) 0
=
𝐵) 1
𝐶)3
𝐷) 𝐷. 𝑁. 𝐸
2𝑥 − 1
lim
=
𝑥→0
𝑥
𝐴) 0.6931471806
B) 0.6921471806
𝐶) ln 2
𝐷) 𝐷. 𝑁. 𝐸
EX.
Floor and ceiling functions
The floor function is the function that takes as input a real number x, and gives as
output the greatest integer less than or equal to x
The ceiling function is the function that takes as input a real number x, and gives
as output the least integer greater than or equal to x
𝐴) 0
B) 1
𝐶) -1
𝐷) 𝐷. 𝑁. 𝐸
Using Factorization and Long division or synthetic division methods
EX.
EX.
Using the conjugate rule
The limit is not unique
Here we set f (x) = sin (π/ x ) and show that the
function can have no limit as x → 0
The function is not defined at x = 0, as you know, that’s
irrelevant. What keeps f from having a limit as x → 0 is
indicated in Figure 2.1.13. As x → 0, f(x) keeps oscillating
between y = 1 and y = –1 and therefore cannot remain close
to any one number L.
The Squeezing Theorem (The Sandwich Theorem)
( x )= 0.
Example: Show that lim x 2 sin 
x→0
Thanks
for listening
WHOA!
Prepare yourself for the upcoming quiz !