Limits Involving Infinity: A Comprehensive guide
MAT1241: Calculus1
University of Eswatini
1. Introduction
When dealing with limits, we encounter situations where the function values become extremely
large or small. These scenarios involve infinity. We’ll focus on two main types of limits:
1. Limits at Infinity: How does a function behave as (x) approaches positive or negative
infinity?
2. Vertical Asymptotes: When does a function approach infinity or negative infinity at
specific points?
2. Limits at Infinity
a. Positive Infinity
Definition
The limit of a function (f(x)) as (x) approaches positive infinity is denoted as:
● If (f(x)) becomes arbitrarily large (positive) as (x) becomes larger and larger, we say that
the limit is .
● Example:
Techniques for Calculating Limits at Positive Infinity
1. Direct Substitution: If the function is continuous, substitute and evaluate.
2. Rational Functions: Compare the degrees of the numerator and denominator.
3. L’Hôpital’s Rule: Useful for indeterminate forms like .
b. Negative Infinity
Definition
The limit of a function (f(x)) as (x) approaches negative infinity is denoted as:
If (f(x)) becomes arbitrarily large (negative) as (x) becomes more negative,
we say that the limit is .
MAT1241: Calculus1
University of Eswatini
1. Introduction
When dealing with limits, we encounter situations where the function values become extremely
large or small. These scenarios involve infinity. We’ll focus on two main types of limits:
1. Limits at Infinity: How does a function behave as (x) approaches positive or negative
infinity?
2. Vertical Asymptotes: When does a function approach infinity or negative infinity at
specific points?
2. Limits at Infinity
a. Positive Infinity
Definition
The limit of a function (f(x)) as (x) approaches positive infinity is denoted as:
● If (f(x)) becomes arbitrarily large (positive) as (x) becomes larger and larger, we say that
the limit is .
● Example:
Techniques for Calculating Limits at Positive Infinity
1. Direct Substitution: If the function is continuous, substitute and evaluate.
2. Rational Functions: Compare the degrees of the numerator and denominator.
3. L’Hôpital’s Rule: Useful for indeterminate forms like .
b. Negative Infinity
Definition
The limit of a function (f(x)) as (x) approaches negative infinity is denoted as:
If (f(x)) becomes arbitrarily large (negative) as (x) becomes more negative,
we say that the limit is .
