Calculus I - Limits and Continuity

I've decided to venture out and take an online course via SAIL. You guessed it, Calculus! I'm excited about this one, so I hope you are too. 

 Limits and Continuity 

Limits:

Do not care about what happens at a point, simply care about what happens as you get infinitely closer to the point.

Methods to find the limit:

1.) Plug in the value the limit approaches. 

2.) If an error occurs, try:

- Factoring

- Mulitiplying by the conjugate

- Foiling or Expanding

- Creating common denominators

- For indeterminant forms (0/0 or inf/inf) use L'Hopital's Rule [below]

L'Hopitals Rule: derive the numerator and denominator (keep it in fractional form), then plug in the value the limit approaches. If after plugging in the value, you come back to the original limit, that means that there may be no proper conclusion. 

Properties of Limits:

Rules for limit as x approaches 0 for sin(x) / x:

Limit as x approaches 0 for (sin(x)/ x) is always 1.

Limit as x approaches 0 for (sin(ax)/ sin(bx)) is always a/b. 

Limit as x approaches 0 for (sin(ax)/bx) is always a/b. 

Limit as x approaches 0 for sin(1/x) is always non-existent. We don't really know what happens at x = 0.

Limits that never exist:

Limit as x approaches 0 (1/x^2). However, some may argue this is infinity. 

Limit as x approaches 0 for sin(1/x)

Limit as x approaches 0 for x/|x| (left and right limits will never 

Continuity

Types:

Removable Discontinuity:

If you have get a zero in the denominator (undefined) and you can use a method to get rid of the "undefined" , then you have a removable discontinuity. This would appear as an open circle on the graph:

Jump Discontinuity:

The left and right side limits exist, and are finite, but aren't equal. You will find the graph approaches the same x-value at two different y-values:

Infinite Discontinuity:

Either one or both the left and right side limits do no exist or are infinite. You will find that one or both sides will go to plus or minus infinity, or it will be terribly hard to figure out what is happening at a point:

Rules for Continuity:

For a function to be continuous at "c":

- f(c) should be defined at c. 

- Limit as x approaches c of f(x) should exist.

- f(c) = limit as x approaches 0 of f(x). 

Intermediate Value Theorem:

At an interval [x1, x2], if the function is continous, the function should be defined (have y-values for all x-values) within that interval.