Basic calculus Gr11

1. BASIC
CALCULUS

2. Unit I
Limits and Continuity

3. REVIEW

4. 1. f(x) = 1 + 3x
Complete the table of values below
given the functions.
x -3 -2 -1 0 1 2 3
f(x) -8 -5 -2 1 4 7 10

5. a. Describe the limit of a function using
correct notation.
b. Estimate the limit of a function or to
identify when the limit does not exist using
a table of values.
c. Estimate the limit of a function or to
identify when the limit does not exist using
OBJECTIVES

6. If f(x) becomes arbitrary close to a
unique number L as x approaches c from
either side, then the limit of f(x) as x
approaches c is L.
Read as:
“the limit of f(x), as x approaches c is L.”
LIMITS
lim
𝑥→𝐶
𝑓 𝑥 = 𝐿

7. The limit of a function 𝒍𝒊𝒎
𝒙→𝑪
𝒇 𝒙 = 𝑳 is
not the same as evaluating a function f(c)
because they are different in terms of
concept. The limit of a function gets its
value by providing inputs that approaches
the particular number while evaluating a
function is more like direct substitution
process.

8. One sided limit
is the value (L) as the x value gets
closer and closer to a certain value c from
one side only (either from the left or from
the right side). In symbols,
lim
𝑥→𝑐+
𝑓 𝑥 = 𝐿 lim
𝑥→𝑐−
𝑓 𝑥 = 𝐿

9. The notion of the limit of a function is
suggested by the question:
“What happens to f(x) as x gets nearer and
nearer to c (but x ≠ 𝐜)?
“Does f(x) approach some number L?”
The question implies that we have to find the
“limit of f(x) as x approaches c, or 𝐥𝐢𝐦
𝒙→𝑪
𝒇(𝒙)

10. A. Direct Substitution
(Try to evaluate function directly)
B. Asymptote (probably)
f(a) =
𝑏
0
where b is not zero
Example: lim
𝑥→1
1
𝑥−1
Calculating lim
𝑥→𝑎
𝑓 𝑥 = 𝐿
Inspect with a graph or
table to learn more about
the function at x = a

11. C. Limit found (probably)
f(a) = b where b is a real number
Example: lim
𝑥→3
𝑥2
= 32
= 9
D. Indeterminate form
f(a) =
𝟎
𝟎
Example: lim
𝑥→−1
𝑥2−𝒙 −𝟐
𝑥2−2𝑥−3

12. Rewriting limit in an equivalent form.
E. Factoring
Example: lim
𝑥→−1
𝑥2−𝒙 −𝟐
𝑥2−2𝑥−3
can be reduced to
lim
𝑥→−1
𝒙 −𝟐
𝑥−3
by factoring and cancelling.

13. F. Conjugates
Example: lim
𝑥→4
𝑥−2
𝑥 − 4
can be written as
lim
𝑥→4
1
𝑥 +2
using conjugates and cancelling.

14. G. Trig Identities
Example: lim
𝑥→0
sin(𝑥)
sin(2𝑥)
can be written as
lim
𝑥→0
1
2cos(𝑥)
using trigonometric identities.

15. H. Approximation
When all else fails, graphs and tables
can help approximate limits.

16. 1. lim
𝑥→2
3𝑥 − 2
Illustrating the Limit of a Function
x 1.9 1.99 1.999 2 2.001 2.01
f(x) 3.700 3.970 3.997 4.003 4.030
?
as x is approaching from the left and from the
right, f(x) is getting closer and closer to 4
∴ lim
𝑥→2
3𝑥 − 2 = 4

17. 2. lim
𝑥→2
1 + 3𝑥
as x is
approaching from
the left and from
the right, f(x) is
getting closer and
closer to 7
∴ 𝒍𝒊𝒎
𝒙→𝟐
𝟏 + 𝟑𝒙 = 7

18. 3. lim
𝑥→−1
(𝑥2
+ 1)
as x is
approaching from
the left and from
the right, f(x) is
getting closer and
closer to 2
∴ 𝒍𝒊𝒎
𝒙→𝟐
𝟏 + 𝟑𝒙 = 2

19. 4. lim
𝑥→2
(𝑥 + 3) solve for its limit when x approaches 2
as x is
approaching from
the left and from
the right, f(x) is
getting closer and
closer to 2
∴ 𝒍𝒊𝒎
𝒙→𝟐
𝒙 + 𝟑 = 2
x<2 f(x)
0
1
1.5
1.9
1.99
1.999
x>2 f(x)
4
3
2.5
2.1
2.01
2.001

20. 5. lim
𝑥→2
(4 + 𝑥) solve for its limit when x approaches 2
as x is
approaching from
the left and from
the right, f(x) is
getting closer and
closer to 2
∴ 𝒍𝒊𝒎
𝒙→𝟐
(𝟒 + 𝒙)=
x<2 f(x)
0
1.5
1.99
1.9999
x>2 f(x)
3
2.5
2.01
2.0001

21. A. Direct Substitution
(Try to evaluate function directly)
B. Asymptote (probably)
f(a) =
𝑏
0
where b is not zero
Example: lim
𝑥→1
1
𝑥−1
Techniques in Calculating Limits
lim
𝑥→𝑎
𝑓 𝑥 = 𝐿
Inspect with a graph or
table to learn more about
the function at x = a

22. A. Direct Substitution
(Try to evaluate function directly)
Different Techniques in Calculating
Limits

23. EXAMP
LE
1. lim
𝑤→1
(1 + 3
𝑤)(2 − 𝑤2
+ 3𝑤3
)
= (1 +
3
1)(2 − 12
+ 3(1)3
)
= (1 + 1)(2 − 1 + 3)
= (2)(4)
= 8
2. lim
𝑡→−2
𝑡2 − 1
𝑡2 + 3𝑡 − 1
=
(−2)2−1
(−2)2+3(−2)−1
=
4−1
4−6−1
=
3
−3
= −1

24. 3. lim
𝑧→2
(
2𝑧 + 𝑧2
𝑧2 + 4
)3
= (
2(2)+ (2)2
(2)2+4
)3
= (
4 + 4
4 + 4
)3
= (
8
8
)3
= 1
4. lim
𝑥→0
𝑥2 − 𝑥 − 2
𝑥3 + 6𝑥2 − 7𝑥 + 1
=
02−0 −2
03+6(0)2−7(0)+1
=
−2
1
= −2

25. Boardwork!
1. lim
𝑦→−2
4 − 3𝑦2 − 𝑦3
6 − 𝑦 − 𝑦2
2. lim
𝑥→−1
𝑥3 − 7𝑥2 + 14𝑥 − 8
2𝑥2 + 3𝑥 − 4
3. lim
𝑥→−1
𝑥2 + 3 − 2
𝑥2 + 1

26. Different Techniques in Calculating
Limits
Rewriting limit in an equivalent form.
B. Factoring
Example: lim
𝑥→−1
𝑥2−𝒙 −𝟐
𝑥2−2𝑥−3
can be reduced to
lim
𝑥→−1
𝒙 −𝟐
𝑥−3
by factoring and cancelling.

27. EXAMPLE
1. lim
𝑥→0
5𝑥
𝑥2 + 2𝑥
=
5(0)
02+2(0)
=
0
0
= 0
Indeterminate
= lim
𝑥→0
5𝑥
𝑥2 + 2𝑥
= lim
𝑥→0
5
(𝑥 + 2)
= lim
𝑥→0
5𝑥
𝑥(𝑥 + 2)
=
5
(0 + 2)
=
5
2

29. Boardwork!
1. lim
𝑥→−2
𝑥2 + 3𝑥 − 10
𝑥 − 2
2. lim
𝑥→3
𝑥2 − 9
𝑥 − 3
3. lim
𝑥→4
4 − 𝑥
𝑥2 − 16

30. Points to remember!
1. Substitute. Plug in the limit value into the
functuon. If the answer is a real number, that is a
limit.
2. If we substitute and we get indeterminate form
0/0, we need to simplify the fraction by

31. Different Techniques in
Calculating Limits
Conjugates
Example: lim
𝑥→4
𝑥−2
𝑥 − 4
can be written as
lim
𝑥→4
1
𝑥 +2
using conjugates and cancelling.

32. Limits that have sum or differences involving
square roots can be simplified by
conjugation