1.1) Can Change Occur in an Instant?
The function b(x) represents specific information, such as the buffalo population in 1890, represented by b(90). The difference between b(x) and b(y) represents the rate of change per x-y, for example, b(50) - b(0) which is the average rate of change of the buffalo population per year from 1800 to 1850. Furthermore, the difference between b(x) and b(x-0.001) gives us an estimate of the instantaneous rate of change at a specific x-value, for example, b(32) - b(31.999) represents the estimate of the instantaneous rate of change of the buffalo population per year at the time 1832.
Limits Graphically
1.2-1.3)
A limit is the y-value a function approaches from both the left and the right side of a given x-value. It is denoted as lim f(x) = L, which means that if f(x) becomes arbitrarily close to a single number as x approaches c from either side, then the limit is L. One-sided limit represents the y-value a function approaches as you approach a given x-value from either side. A limit does not exist when the left and right-handed limits are different, there is oscillating behavior at the x-value, or unbounded behavior (∞0). Common examples and solutions for trigonometric limits and squeeze theorem are also a part of this concept.
Finding Limits From Tables
1.4)
Algebraic Properties of Limits and Piecewise Functions are important in evaluating limits using algebraic techniques. Understanding properties like x + x = 2x and all other properties of limits, and applying them to piecewise functions is crucial for finding limits from tables.
Basic Limits
1.5)
The basic limits and identity rules such as lim (x+c) = L, lim x^n = c^n, need to be understood thoroughly to solve limits involving algebraic manipulation.
Algebraic Manipulation (Limits)
1.6)
Special trigonometric limits and trigonometric functions are to be taken into account when manipulating algebraic limits as they need to approach 0. Also, using notation and writing "limit" correctly can be essential during this manipulation.
The Squeeze Theorem
1.8)
The Squeeze Theorem, also known as the Sandwich Theorem or Pinching Theorem, is an important concept in finding limits when a function is squeezed between two other functions, and all three functions are approaching the same limit.
Types of Discontinuities
1.10)
Understanding the types of discontinuities such as holes, vertical asymptotes, and jump discontinuities, is crucial for analyzing the behavior of functions at specific points.
Multiple Representations (Limits)
1.9)
Different representations of functions and limits need to be studied to grasp the various ways limits can be represented and understood.
Defining Continuity at a Point
1.11)
The continuity at a point needs to be defined by ensuring that f(c) is defined, lim f(x) exists, and f(x) is defined.
Confirming Continuity Over an interval
1.12)
Apart from defining continuity at a point, it is also crucial to confirm continuity over an interval by considering the restrictions in the domain and understanding the behavior of functions within that interval.
Removing Discontinuities
1.13)
Removing discontinuities is a part of analyzing functions and limits, and it involves steps of defining continuity to ensure a coherent and smooth function.
Infinite Limits and Vertical Asymptotes
1.14)
Understanding infinite limits and vertical asymptotes is essential as they play a significant role in the behavior of a function as it approaches infinity or negative infinity.
Limits at Infinity and Horizontal Asymptotes
1.15)
Having a clear understanding of limits at infinity and horizontal asymptotes is essential for analyzing the end behavior of a function and its approach towards infinity.
Intermediate Value Theorem (INT)
1.16)
The Intermediate Value Theorem states that if a function is continuous on a closed interval and takes on two distinct values, then it must also take on every value in between. This concept is crucial in understanding the continuity of functions over a given interval.
This cheat sheet encompasses various algebraic properties of limits and their applications, along with examples and solutions to help understand and apply these concepts effectively.
