Exploring the concept of limits at infinity in calculus, this overview discusses how functions behave as input values, represented by x, approach positive or negative infinity. It covers the formal definition, graphical analysis, horizontal asymptotes, and the application of mathematical rules like the Squeeze Theorem to compute these limits. The text also highlights the importance of horizontal asymptotes as graphical representations of finite limits and the use of various theorems to evaluate function behavior in the long term.
Show More
Limits at infinity describe the behavior of functions as their input values increase or decrease without bound
The formal definition states that a function approaches a limit as its input approaches infinity if, for every positive number, there is a corresponding number such that the function's values are within a certain distance from the limit
Limits at infinity are denoted by the symbols +∞ or -∞, representing positive or negative infinity, respectively
Graphical analysis involves plotting the function and horizontal lines to determine if the function's values fall within certain bounds for large input values
The function e^-x + 1 illustrates that as x becomes very large, the function's values get arbitrarily close to 1, indicating a limit of 1 at infinity
The function √x grows without bound as x increases, indicating an infinite limit at infinity
The Sum, Difference, Product, Constant Multiple, and Quotient Rules are extensions of the Limit Laws for finite limits
When both the numerator and denominator of a function approach infinity, the Quotient Rule requires algebraic manipulation to be applicable
The Squeeze Theorem states that if a function is sandwiched between two other functions that both approach the same limit at infinity, then the function must also converge to that limit
Limits at negative infinity follow similar principles as limits at positive infinity, with the condition that the input values are very negative
Graphical analysis can also be used to evaluate limits at negative infinity
It is important to interpret graphs and tables with caution, as they can sometimes be misleading
00
Definition of limit at negative infinity
Behavior of function as x approaches negative infinity; x becomes very negative.
01
Graphical analysis for limits at negative infinity
Use of graph to observe function's behavior as x decreases without bound; look for approaching value.
02
Function example: f(x) = 1/x * sin(x)
As x approaches negative infinity, f(x) oscillates but amplitude diminishes, approaching zero.
