This content delves into the calculus concepts of limits at infinity and asymptotes, explaining how they reveal the behavior of functions as inputs grow large. It covers the identification of horizontal, vertical, and slant asymptotes, each indicating different trends in a function's growth or decay. Understanding these concepts is essential for analyzing functions and their graphs over extensive domains, providing insights into their behavior at extreme values.
Show More
Limits at infinity describe the behavior of functions as the input values become very large in magnitude
Trend of Function Values
When evaluating limits at infinity, one seeks to understand the trend of the function's values as the variable approaches positive or negative infinity
Divergence and Convergence
Limits at infinity help us determine whether a function diverges or converges to a particular value
Exponential and logarithmic functions are common examples where limits at infinity are analyzed to understand their asymptotic behavior
Asymptotes are lines that a graph of a function approaches but never reaches as the inputs or outputs extend towards infinity
Horizontal Asymptotes
Horizontal asymptotes indicate that a function approaches a constant value as the input becomes very large or very small
Vertical Asymptotes
Vertical asymptotes occur when a function's value becomes unbounded at certain finite input values
Slant Asymptotes
Slant asymptotes are found when the degree of the numerator of a rational function exceeds the degree of the denominator by one
Horizontal Asymptotes
Horizontal asymptotes are determined by comparing the degrees of the numerator and denominator polynomials in a rational function
Vertical Asymptotes
Vertical asymptotes are found by identifying the values for which the function becomes undefined
Slant Asymptotes
To find the equation of a slant asymptote, one must divide the function by the input variable and calculate the limit as the input approaches infinity
A thorough understanding of limits at infinity and asymptotes is crucial for analyzing the behavior of functions over large domains
Asymptotes act as boundaries that a function may approach but not cross, offering insights into the function's behavior at extreme values
Horizontal, vertical, and slant asymptotes each have distinct identification criteria and implications for the function's graph
00
A function ______ when its values rise or fall indefinitely, as opposed to converging to a specific value.
diverges
01
______ and ______ functions often have their asymptotic behavior analyzed through limits at infinity.
Exponential
logarithmic
02
Define Asymptotes
Lines a graph approaches but never reaches, as inputs/outputs head towards infinity.
