1. Limits, Continuity, Basic Differentiability, and Computing Derivatives
By: Sameer, Snigdha, Aditya

2. Limits
Recall that…
A limit is when a function gets super close to a number from both sides of x, but the function never reaches that number…
It’s predicting a number between two neighboring points.
Let’s think about how we can solve these

3. Solving Limits - Ex1: Make a table
-0.1
-.01
-.001
0.001
0.01
0.1 Y
Plug in on your calculator!

4. Solving Limits - Ex1: Make a table (cont.)
-0.1
-.01
-.001
0.001
0.01
0.1 Y
0.998
0.999 1
Therefore, the answer is 1 because as x approaches 0 from both sides, the value of the limit gets closer and closer to 1.

5. Solving Limits - Ex2: Graph
As x approaches 0 from both sides, what does f (x) get arbitrarily close to?
lim f(x) = 2
x->0

6. Solving Limits - Ex3: Direct Substitution
When a function is given, your first tactic to solving a limit should be plugging in the value the limit approaches.
Plug in x = 5!

7. Solving Limits - Ex4: Factoring/Simplifying
If direct substitution yields an indeterminate answer (0/0 or infinity/infinity), try factoring!

8. Solving Limits - Ex4: Factoring/Simplifying
Let’s try factoring....
And now let’s simplify...
Now when we use direct substitution, we get 8/2, which is 4

9. Solving Limits - Ex5: Rationalizing the Numerator/Denominator
When you can’t use direct substitution or factoring, check for a radical and try rationalizing the numerator or denominator

10. Solving Limits - Ex5: Rationalizing the Numerator/Denominator
Multiply either the numerator or denominator by a radical that will get rid of the radical sign and use a different sign for the second term…

11. Solving Limits - Ex5: Rationalizing the Numerator/Denominator
Then, simplify:

12. Solving Limits - Ex5: Rationalizing the Numerator/Denominator
Finally, use direct substitution:

13. Limits - Does Not Exist (DNE)
This limit does not exist because on the limit from the left of x approaches 1 while on the right side of x it approaches 2.

14. Limits - Vertical Asymptotes
If f(x) approaches infinity or negative infinity as x approaches c from the right or left, then there is a vertical asymptote at x = c
lim f(x) = infinity; therefore VA
x->0

15. Limits - Vertical Asymptotes (cont.)
A constant divided by a small number yields infinity:
This indicates a potential vertical asymptote

16. Limits - Horizontal Asymptote
The line y = L is a horizontal tangent if limit as x approaches infinity for f(x) = L

17. Limits - Horizontal Asymptotes (cont.)
Use coefficient rule: if highest power in the numerator and denominator match, then divide the coefficients for the limit/horizontal asymptote
=3/2
If the power is smaller in numerator or the numerator has a slower function then the limit goes to 0.

18. Continuity
Consider this hole in the road...is the road continuous?

19. Continuity - Definition
Most of you probably agreed that the road is not continuous...you wouldn’t want to drive over that!
But...what would make the road continuous?
Well, the road would have to:
-Fill the hole (discontinuity filled by f(c))
-There would have to be road on either side of the road (limit must exist)
-The road on either side and the filled in hole would have to connect (limit must equal f(c)

20. Continuity - Definition (cont.)
Similarly in calculus these rules also apply!
The limit as x approaches c for f(x) must exist
f(c) must exist and have a value
The limit as x approaches c for f(x) must equal f(c)
Now, justify if each of the points are continuous or not.

21. Continuity - Justification
Continuous.The limit as x approaches 0 is 1 and f(0) is also 1
Discontinuous. The limit as x approaches -2 does not exist
Discontinuous. The limit as x approaches 3 is 0, but f(3) is -1. Therefore there is a discontinuity at x = 3

22. Continuity - Practice Problem
Hmm...let’s use the definition of continuity to solve this problem

23. Continuity - Practice Problem (cont.)
First, let’s look at x=1. f(1) = 2 so the limit as x approaches 1 must also equal 2. Use direct substitution!

24. Continuity - Practice Problem (cont.)
Plugging in x=1 gives us a + b = 2
Now let’s look at x = 3. f(3) gives -2 so the limit as x approaches 3 must also be -2. Again, use direct substitution…
Plugging in x=3 gives us 3a + b = -2

25. Continuity - Practice Problem (cont.)
Now, setup a system of equations:
a + b = 2
3a + b = -2
Solving for a gives - 2a = 4
a = -2. Plugging back in, we get
-2 + b = 2, and
b = 4

26. Basic Differentiability
A function is differentiable if its derivative exists at every point on its domain.
A derivative is essentially the slope of a function and is denoted as f’(x)
Differentiability implies Continuity
However, continuity does not mean a function is 100% differentiable

27. Basic Differentiability - Non-differentiable functions
Corner
Cusp

28. Basic Differentiability - Non-differentiable functions (cont.)
Vertical Tangent
Jump Discontinuity

29. Computing a Derivative
Formal Definition of a Derivative
This simply means the derivative of cos(x) at x= pi

30. Computing a Derivative - Polynomials
Here’s the formula for computing a derivative*
*Note: there are special ways to derive trig, logarithmic, e^x, and other special case functions

31. Computing a Derivative - Polynomials (cont.)

32. Computing a Derivative - Rational Functions
Quotient Rule: When computing functions with quotients, you must use quotient rule.

33. Computing a Derivative - Rational Functions (cont.)
Use quotient rule!

34. Computing a Derivative - Rational Functions (cont.)
Derivative of f(x)!

35. Computing a Derivative - Trigonometric Functions
Memorize these Trig Derivatives!

36. Computing a Derivative - Trigonometric Functions (cont.)
Quick introduction to chain rule: a derivative rule in which nested functions must be included in the derivative.

37. Computing a Derivative - Trigonometric Functions (cont.)
h(x) = sin(2x)...Find the derivative
f(x) = sin(x)
g(x) = 2x
so:
h’(x) = cos(2x)*2 or 2cos(2x)

38. Computing a Derivative - Exponential/Logarithmic Functions
You should know these derivatives:

39. Computing a Derivative - Exponential/Logarithmic Functions (cont.)
Finally, product rule:`

40. Computing a Derivative - Exponential/Logarithmic Functions (cont.)
Let’s take the derivative of this!
Using chain rule, we get:

41. That’s All Folks…
FIN.