SECTIONS ## Introduction to Limits

The limit only exists if you can approach it from both sides

We know that the domain of any logarithm is only the positive numbers and so we can’t even talk about the left-handed limit because that would necessitate the use of negative numbers. Likewise, since we can’t deal with the left-handed limit then we can’t talk about the normal limit.

The limit operator forms some kind of group over the reals

FAQ

How do I know that this function has two solutions?

g(x)=x+7/x^2−4

Because there is an X ^2 in the denominator. ### √(x^2)=|x| Don’t forget that √(x^2)=|x|

Computing Limits from a Graph 5:38

[ _{{x ^-}} f(x) = 1 ]

[ _{{x ^+}} f(x) = 4 ]

[ _{{x }} f(x) ] The limit only exists if you can approach it from both sides ## Computing Basic Limits by plugging in numbers and factoring 8:37 The first rule is if you plug a limit into a function and You get an answer you’re done

When algebraically manipulating limits, Continue writing the limit operator until you take the limit.

Problem-Solving Strategy P F G O Plug in, factor, get common denominator,open parenthesis ### Calculating a Limit When f(x)/g(x) has the Indeterminate Form 0/0 First, we need to make sure that our function has the appropriate form and cannot be evaluated immediately using the limit laws. We then need to find a function that is equal to h(x)=f(x)/g(x) for all x≠a over some interval containing a. To do this, we may need to try one or more of the following steps: If f(x) and g(x) are polynomials, we should factor each function and cancel out any common factors. If the numerator or denominator contains a difference involving a square root, we should try multiplying the numerator and denominator by the conjugate of the expression involving the square root. If f(x)/g(x) is a complex fraction, we begin by simplifying it. Last, we apply the limit laws.

Limit using the Difference of Cubes Formula 1 12:34

Limit with Absolute Value 14:53

Limit by Rationalizing 16:29

Limit of a Piecewise Function 19:34

Trig Function Limit Example 1 21:37

Trig Function Limit Example 2 22:38

Trig Function Limit Example 3 23:41

Continuity 25:35

Removable and Nonremovable Discontinuities 34:06

Intermediate Value Theorem 42:42

Infinite Limits 49:4

The function f(x) will have a horizontal asymptote at y=L

if either of the following are true. limx→∞f(x)=Llimx→−∞f(x)=L ### Notes If you try to take the limit of a function And while expanding the accuracy the result is unbounded then it is infinity. The sign of Infinity is inherited from the sign of the result of the function as you try to take the limit.

It is useful to point out that functions of the form f(x)=1/(x−a)^n, where n is a positive integer, have infinite limits as x approaches a from either the left or right

Infinite Limits from the Left

Let f(x) be a function defined at all values in an open interval of the form (b, a).

Infinite Limits from the Right

Let f(x) be a function defined at all values in an open interval of the form (a, c).

Two-Sided Infinite Limit

Let f(x) be defined for all x ≠ a in an open interval containing a.

Infinite Limits from Positive Integers

Bottom heavy rational limits equate to zero.

Equal rational limits will be the ratio of their leading coefficients.

For top heavy rational limits evaluate the limit at Infinity accordingly ## Vertical Asymptotes 57:33

Vertical Asymptotes Definition

Let f(x) be a function. If any of the following conditions hold, then the line x = a is a vertical asymptote of f(x):

Derivative

Infinitesimal Derivatives

Ah, now I see what you’re asking! You’re referring to an infinitesimal approach to derivatives, where we think of changes in ( y ) and ( x ) as small increments and then find the ratio of their changes. This is a more intuitive or “algebraic” method that focuses on how small changes in one variable relate to small changes in the other.

Let’s go through the process using this idea!

The Infinitesimal Approach

The idea behind this approach is that we can make a small change in ( x ) (denoted by ( x )) and observe the corresponding small change in ( y ) (denoted by ( y )).

Then the derivative ( ) can be thought of as the ratio of these small changes in ( y ) and ( x ) as ( x ) becomes very small.

The steps look like this:

  1. Start with a function: Let’s use ( y = f(x) = x^2 ) again as an example.

  2. Add a small change ( x ) to ( x ): This gives us the new value of the function at ( x + x ).

[ f(x + x) = (x + x)^2 ]

  1. Find the corresponding change in ( y ): The change in ( y ), denoted by ( y ), is:

[ y = f(x + x) - f(x) = (x + x)^2 - x^2 ]

  1. Expand and simplify: Expand ( (x + x)^2 ) to get:

[ (x + x)^2 = x^2 + 2xx + (x)^2 ]

Now, subtract ( f(x) = x^2 ):

[ y = x^2 + 2xx + (x)^2 - x^2 ]

The ( x^2 ) terms cancel out, leaving us with:

[ y = 2xx + (x)^2 ]

  1. Form the ratio ( ): Now, we can express the change in ( y ) divided by the change in ( x ), which gives us:

[ = ]

  1. Simplify the ratio: Factor out ( x ) from the numerator:

[ = 2x + x ]

  1. Take the limit as ( x ): As ( x ) becomes smaller and smaller (infinitesimally small), the term ( x ) itself approaches 0. So, in the limit, we get:

[ _{x } = 2x ]

Thus, we arrive at the derivative:

[ = 2x ]

Intuition

In this process, ( y ) is the change in ( y ) when ( x ) changes by a small amount ( x ). The derivative ( ) tells us the rate at which ( y ) changes with respect to ( x ), and as ( x ) becomes very small, the ratio ( ) gives the exact slope of the function at a particular point.

General Form

In general, for any function ( y = f(x) ), we can express the derivative as:

[ = _{x } ]

This approach mirrors what you were describing, where we think of small increments in ( y ) and ( x ) and then take the ratio of those increments to find the derivative.

Does this match the algebraic process you were thinking of? Let me know if you’d like more clarification or examples!

Definition of the Derivative Example 1:12:10

Derivative Formulas 1:15:39

More Derivative Formulas 1:23:11

Product Rule 1:26:59

Quotient Rule 1:30:26

Chain Rule 1:34:10

Average and Instantaneous Rate of Change (Full Derivation) 1:39:52

Average and Instantaneous Rate of Change (Example) 1:47:37

Position, Velocity, Acceleration, and Speed (Full Derivation) 1:51:42

Position, Velocity, Acceleration, and Speed (Example) 1:59:36

Implicit versus Explicit Differentiation 2:09:25

Critical Numbers 2:18:46

Extreme Value Theorem 2:28:36

Rolle’s Theorem 2:34:50

The Mean Value Theorem 2:41:20

Increasing and Decreasing Functions using the First Derivative 2:48:13

The First Derivative Test 2:53:28

Concavity, Inflection Points, and the Second Derivative 3:00:53

The Second Derivative Test for Relative Extrema 3:10:06

Limits at Infinity 3:14:26

Newton’s Method 3:17:16

Differentials: Deltay and dy 3:21:39

Indefinite Integration (theory) 3:24:11

Indefinite Integration (formulas) 3:34:52

Integral Example 3:41:54

Integral with u substitution Example 1 3:43:21

Integral with u substitution Example 2 3:45:31

Integral with u substitution Example 3 3:47:49

Summation Formulas 3:49:38

Definite Integral (Complete Construction via Riemann Sums) 3:58:04

Definite Integral using Limit Definition Example 4:10:16

Fundamental Theorem of Calculus 4:18:18

Definite Integral with u substitution 4:25:47

Mean Value Theorem for Integrals and Average Value of a Function 4:28:05

Extended Fundamental Theorem of Calculus (Better than 2nd FTC) 4:36:16

Simpson’s Rule 4:44:59 error here: forgot to cube the (3/2) here at the end, otherwise ok!

The Natural Logarithm ln(x) Definition and Derivative 4:50:25

Integral formulas for 1/x, tan(x), cot(x), csc(x), sec(x), csc(x) 4:56:30

Derivative of e^x and it’s Proof 5:02:19

Derivatives and Integrals for Bases other than e 5:08:33

Integration Example 1 5:15:37

Integration Example 2 5:17:16

Derivative Example 1 5:19:15

Derivative Example 2 5:20:32

Properties

Properties of the Exponential Function f(x) = b^x:

  1. Initial Value:

    • f(0) = 1
    • The function will always take the value of 1 at x = 0.

  2. Non-Zero Values:

    • f(x) ≠ 0
    • An exponential function will never be zero.

  3. Positivity:

    • f(x) > 0
    • An exponential function is always positive.
    • Therefore, the range of an exponential function is (0, ∞).

  4. Domain:

    • The domain of an exponential function is (-∞, ∞).
    • In other words, you can plug every x into an exponential function.

  5. Behavior Based on Base b in f(x) = b^x:

    • If 0 < b < 1:
      • f(x) → 0 as x → ∞
      • f(x) → ∞ as x → -∞
    • If b > 1:
      • f(x) → ∞ as x → ∞
      • f(x) → 0 as x → -∞

The natural exponential function f(x) = e^x:

Definition:

The natural exponential function is defined as f(x) = e^x, where e is approximately 2.71828182845905....

Behavior of the Natural Exponential Function:

Properties polynomials limits:

If p(x) is a polynomial, then the limit of p(x) as x approaches a is simply p(a). In other words:

lim (x → a) p(x) = p(a)

This means that for polynomials, you can find the limit by directly substituting a into the polynomial.

tricks

https://www.youtube.com/watch?v=MnGfA2uO6C8 https://tutorial.math.lamar.edu/Problems/CalcI/ComputingLimits.aspx https://openstax.org/books/calculus-volume-1/pages/2-3-the-limit-laws