Limits, Continuity, and Derivatives for functions, Compiled note for easier to understand as well solving for L

4. Solving Limits

5. Graph

6. Solving Limits

7. Graph

8. Solving Limits

9. Graph

10. Solving Limits

11. graph

12. Solving Limits

13. Graph

14. Solving Limits

15. Graph

17. Limit of a Constant
The limit of a constant is itself.
 𝒍𝒊𝒎
𝒙→𝒂
𝒃 = 𝒃

18. Substitution Law
 X is substituted by the value X
approaches.
 𝒍𝒊𝒎
𝒙→𝒎
𝒙 = 𝒎
 In this case, the limit of x as x approaches
m is m.

19. Constant Multiple Theorem

The limit of a multiple of a function is
simply that multiple of the limit of the
function.

20. Addition Theorem
 The limit of a sum of functions is the sum
of the limits of the individual functions and
the limit of a difference of functions is the
difference of their limits.

21. Multiplication Theorem
• The limit of a product of functions is equal
to the product of their limits

22. Division Theorem
 The limit of a quotient of functions is equal to
the quotient of the limits of the individual
functions, provided the denominator limit is
not equal to 0.

23. Power Theorem
The limit of an integer power p
of a function is just that power
of the limit of the function.

24. Radical Theorem
 If n is a positive integer, the limit of the nth
root of a function is just the nth root of the limit
of the function, provided the nth root of the
limit is a real number.
 Thus, it is important to keep in mind that if n is
even, the limit of the function must be
positive.

27. Continuous Function
(graph)

28. Discontinuous Function
(graph)

29. Jump Essential Discontinuity

30. Infinite Essential Discontinuity

31. Removable (Point)
Discontinuity

32. Continuity in Real Life

34. Tangent Lines

36. Secant Lines

37. Solving for the equation of
Tangent Lines

38. Derivatives

40. The Constant Rule
The derivative of f(x) = c
where c is a constant is given
by
f '(x) = 0

41. The Power Rule
The derivative of f(x) = 𝑥 𝑟
where r
is a constant real number is given
by
f '(x) = r∗ 𝒙 𝒓−𝟏

42. The Constant Multiple Rule
The derivative of f(x) = c g(x) is
given by
f '(x) = c∗g'(x)

43. The Sum and Difference Rule
The derivative of f(x) = g(x) + h(x) is
given by
f '(x) = g '(x) + h '(x)
The derivative of f(x) = g(x) - h(x) is
given by
f '(x) = g '(x) - h '(x)

44. The Product Rule
The derivative of f(x) = g(x) h(x) is
given by
f '(x) = g(x) h '(x) + h(x) g '(x)

45. The Quotient Rule
The derivative of f(x) = g(x) /
h(x) is given by
f '(x) =
𝐡 𝐱 𝐠′(𝐱) − 𝐠 𝐱 𝐡′(𝐱)
𝒉(𝒙) 𝟐

47. 1.AUTOMOBILES
 In a automobile there is always an odometer and
a speedometer. These 2 gauges work in tandem
and allow to drive to determine his speed and his
distance that he has traveled. Electronic
versionsof there gauges simply use derivatives
to transform the data sent to the tiresto miles pen
Hour (MPH) and distance (KM).

48. 2. BUSINESS
 In the business world application is when the
data has been charted on graph or data table
such as excel. Once it has been input,the
data can be graphed and with the applications
of derivatives you can estimate the profit and
loss point for certain ventures.

49. 3.CLIMATE CHANGE
In measuring how quickly the
temperature is changing or recaps
are melting, scientist can
determine the severity of various
situations.

50. 4.BIOLOGY