Limits
Limits are all about approaching.And the entire Calculus is built upon this concept.
Refer to Khan academy.
Limits properties
Refer to Khan academy: Limit properties
Limits at infinity
No matter why kinds of Limits you’re looking for, to understand it better, the best way is to read the Step-by-Step Solution from Symbolab:Limit Calculator from Symbolab.)
Step-by-Step Solution
Symbolab
All types of discontinuities
Refer to Mathwarehouse: What are the types of Discontinuities?
Jump Discontinuities
A Jump discontinuity occurs at some point, the left side limit is DIFFERNT with the right side limit.
Jump discontinuity
left side limit
right side limit
Analyzing functions for discontinuities: algebraic
If the limits of both side of some point, are EQUAL, then it’s continuous at this point.
At a point a, for f(x) to be continuous at x=a, we need lim(x→a)f(x) = f(a).
a
f(x)
x=a
lim(x→a)f(x) = f(a)
Solve:It’s just the same with calculating the limits of both sides.
Derivative Basics
Simply saying, it’s just the SLOPE of ONE POINT of a graph (line or curves or anything).
Refer to Mathsisfun: Introduction to Derivatives
Differentiability
“If the point of a function IS differentiable, then it MUST BE continuous at the point.”
Example of NOT differentiable points:
differentiable
Derivative equation
The idea of derivative equation is quite simple: The LIMIT of the SLOPE.
The slope is equal to change in Y / change in X. So for a point a, we IMAGINE we have another near point which lies on the SAME LINE with a, and since we have TWO POINTS now, we can then let…
change in Y / change in X
Local linearity & Linear approximation
Local linearity is for approximating of a point's value by its near known point.
Local linearity
Just think of a curve, a good way to approximate its Y-value, is to find another known point near it, and make a line connecting two points, then gets the…
