Key Concepts
● Limits describe the behavior of a function as the input approaches a certain value
● Continuity refers to a function being defined at every point within its domain
without any breaks or gaps
● One-sided limits consider the function's behavior as the input approaches a value
from either the left or right side
● Infinite limits occur when the output of a function grows arbitrarily large or small
as the input approaches a certain value
○ Vertical asymptotes are associated with infinite limits and represent a line
that the function approaches but never reaches
● Limit laws and properties enable the evaluation and simplification of complex limit
expressions
● Applications of limits include analyzing the behavior of functions in real-world
scenarios and solving optimization problems
Limit Definition and Notation
● The limit of a function f(x)f(x) as xx approaches a value aa is denoted as
limx→af(x)=Llimx→af(x)=L
● This notation means that as xx gets closer to aa (but not necessarily equal to aa),
the output f(x)f(x) gets arbitrarily close to LL
● The limit does not depend on the function's value at x=ax=a, but rather the
behavior of the function near aa
● Limits can be evaluated from both the left and right sides of aa, denoted as
limx→a−f(x)limx→a−f(x) and limx→a+f(x)limx→a+f(x), respectively
● For a limit to exist, the left-hand and right-hand limits must be equal
● The limit of a function can exist even if the function is undefined at the point of
interest
Evaluating Limits
● Direct substitution can be used to evaluate limits when the function is continuous
at the point of interest
○ Simply substitute the value of aa into the function f(x)f(x) to find the limit
● Limits describe the behavior of a function as the input approaches a certain value
● Continuity refers to a function being defined at every point within its domain
without any breaks or gaps
● One-sided limits consider the function's behavior as the input approaches a value
from either the left or right side
● Infinite limits occur when the output of a function grows arbitrarily large or small
as the input approaches a certain value
○ Vertical asymptotes are associated with infinite limits and represent a line
that the function approaches but never reaches
● Limit laws and properties enable the evaluation and simplification of complex limit
expressions
● Applications of limits include analyzing the behavior of functions in real-world
scenarios and solving optimization problems
Limit Definition and Notation
● The limit of a function f(x)f(x) as xx approaches a value aa is denoted as
limx→af(x)=Llimx→af(x)=L
● This notation means that as xx gets closer to aa (but not necessarily equal to aa),
the output f(x)f(x) gets arbitrarily close to LL
● The limit does not depend on the function's value at x=ax=a, but rather the
behavior of the function near aa
● Limits can be evaluated from both the left and right sides of aa, denoted as
limx→a−f(x)limx→a−f(x) and limx→a+f(x)limx→a+f(x), respectively
● For a limit to exist, the left-hand and right-hand limits must be equal
● The limit of a function can exist even if the function is undefined at the point of
interest
Evaluating Limits
● Direct substitution can be used to evaluate limits when the function is continuous
at the point of interest
○ Simply substitute the value of aa into the function f(x)f(x) to find the limit
