Glossary

2.4 Infinite limits and limits at infinity

2 min read•july 22, 2024

Infinite limits and limits at infinity are crucial concepts in calculus. They help us understand how functions behave as they approach specific values or grow without bound. These ideas are key to grasping function behavior and asymptotes.

is a powerful tool for evaluating tricky limits. It's especially useful for , which often pop up when dealing with . Understanding these concepts helps us analyze function and identify asymptotes.

Infinite Limits and Limits at Infinity

Classification of infinite limits

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  • Occur when limit of function as x approaches specific value is infinity or negative infinity
    • Right-hand lim⁡x→a+f(x)=∞\lim_{x \to a^+} f(x) = \infty or −∞-\infty (function approaches infinity as x approaches a from the right)
    • Left-hand infinite limit lim⁡x→a−f(x)=∞\lim_{x \to a^-} f(x) = \infty or −∞-\infty (function approaches infinity as x approaches a from the left)
  • Vertical asymptotes exist at x-values where function approaches infinity or negative infinity
    • Identified by finding x-values that make denominator of rational function equal to zero (division by zero)

Function behavior at infinity

  • Describe behavior of function as x approaches positive or negative infinity
    • lim⁡x→∞f(x)=L\lim_{x \to \infty} f(x) = L as x increases without bound, f(x) approaches value L
    • lim⁡x→−∞f(x)=L\lim_{x \to -\infty} f(x) = L as x decreases without bound, f(x) approaches value L
  • Horizontal asymptotes occur when function approaches specific y-value as x approaches positive or negative infinity
    • For rational functions p(x)q(x)\frac{p(x)}{q(x)}, where p(x) and q(x) are polynomials:
      1. If degree of p(x) < degree of q(x), is y = 0
      2. If degree of p(x) = degree of q(x), horizontal is y = anbm\frac{a_n}{b_m}, where ana_n and bmb_m are leading coefficients of p(x) and q(x)
      3. If degree of p(x) > degree of q(x), no horizontal asymptote exists

L'Hôpital's Rule for indeterminate forms

  • Evaluate limits resulting in indeterminate forms: 00\frac{0}{0}, ∞∞\frac{\infty}{\infty}, 0⋅∞0 \cdot \infty, ∞−∞\infty - \infty, 000^0, 1∞1^\infty, or ∞0\infty^0
  • Apply L'Hôpital's Rule:
    1. Differentiate numerator and denominator separately
    2. Evaluate limit of new quotient
    3. Repeat process if new limit is still indeterminate form
  • Applicable to and limits at infinity

End behavior of rational functions

  • Refers to behavior of function as x approaches positive or negative infinity
  • For rational functions:
    1. Determine degree of numerator and denominator polynomials
    2. Compare degrees to identify presence and value of horizontal asymptotes
    3. Evaluate limits at positive and negative infinity to confirm end behavior
  • Oblique asymptotes occur when degree of numerator is exactly one more than degree of denominator
    • Slant line that function approaches as x approaches positive or negative infinity
    • Find equation of by dividing numerator by denominator using long division and ignoring remainder

Key Terms to Review (22)

Approaching Behavior: Approaching behavior refers to how a function behaves as it nears a specific point or value, particularly in the context of limits. This concept is essential when analyzing infinite limits and limits at infinity, as it helps determine what value, if any, a function is tending toward, especially when the function does not actually reach that value. Understanding this behavior provides insights into continuity, discontinuity, and the overall behavior of functions near critical points.
Arrow: The arrow symbol '→' represents the concept of limits in calculus, indicating the approach of a variable towards a specific value. This notation is essential for expressing limits and helps in visualizing how functions behave near particular points, especially when discussing infinite limits, one-sided limits, and evaluating limits using various techniques.
Asymptote: An asymptote is a line that a curve approaches as it heads towards infinity or a specific point, but never actually touches. This concept is crucial in understanding the behavior of functions at extreme values or around points where they may become undefined. Asymptotes can be horizontal, vertical, or oblique, and they reveal significant information about the limits and behavior of functions, especially when evaluating infinite limits and limits at infinity.
Boundedness: Boundedness refers to the property of a function where its output values remain confined within a specific range, meaning that there exists a real number that serves as both an upper and a lower limit for those values. This concept is crucial when discussing the behavior of functions, particularly in relation to continuity, limits, and optimization, as it helps determine whether functions exhibit certain characteristics over given intervals.
End Behavior: End behavior refers to the way the values of a function behave as the input approaches positive or negative infinity. Understanding end behavior is crucial for analyzing functions, especially when determining horizontal asymptotes and the overall shape of their graphs. It gives insight into the limits of a function at infinity, which can help predict how a function will behave in extreme cases.
Exponential Functions: Exponential functions are mathematical functions of the form $$f(x) = a imes b^{x}$$, where $$a$$ is a constant, $$b$$ is a positive real number, and $$x$$ is the exponent. They describe processes that grow or decay at a constant rate proportional to their current value, making them crucial in modeling real-world phenomena such as population growth and radioactive decay.
Horizontal Asymptote: A horizontal asymptote is a horizontal line that a function approaches as the input values either increase or decrease towards infinity. It helps to describe the end behavior of functions, particularly rational functions, indicating the value that the function approaches but may never actually reach. Understanding horizontal asymptotes is crucial for analyzing limits at infinity and characterizing different types of functions based on their growth or decay patterns.
Indeterminate Forms: Indeterminate forms occur when evaluating limits leads to an ambiguous result that doesn't provide enough information to determine the limit's value. These forms often arise in calculus, particularly in the context of infinite limits and limits at infinity, and are crucial for applying specific techniques like L'Hôpital's Rule to resolve them and find meaningful limit values.
Infinite limit: An infinite limit occurs when the value of a function grows without bound as the input approaches a certain point. This concept is crucial for understanding behaviors of functions near vertical asymptotes and helps in analyzing limits at infinity, where the function approaches infinity as the input increases or decreases indefinitely. Recognizing infinite limits aids in grasping the broader ideas of continuity and discontinuity in functions.
L'Hôpital's Rule: L'Hôpital's Rule is a mathematical method used to evaluate limits that result in indeterminate forms, typically when direct substitution yields results like 0/0 or ∞/∞. This rule states that for functions f(x) and g(x) that are differentiable in a neighborhood around a point (except possibly at the point itself), if both f(x) and g(x) approach 0 or ±∞, the limit of their quotient can be found by taking the limit of the derivatives of these functions instead.
Lim: In calculus, 'lim' refers to the limit of a function as it approaches a specific point or infinity. Understanding limits is crucial because they help us analyze the behavior of functions at points where they might not be explicitly defined, such as points of discontinuity or at infinity. The concept of limits forms the backbone for more advanced topics, including derivatives and integrals, as it allows us to rigorously define these operations in a mathematical sense.
Lim (x→-∞) f(x) = l: The expression lim (x→-∞) f(x) = l indicates that as the variable x approaches negative infinity, the function f(x) approaches a specific value l. This concept is key in understanding how functions behave at the extremes of their domains, particularly when looking at limits at infinity and infinite limits. It helps in analyzing the end behavior of functions and can provide insights into asymptotic behavior and the overall shape of the graph of a function.
Lim (x→∞) f(x) = l: The notation lim (x→∞) f(x) = l describes the limit of a function f(x) as x approaches infinity, indicating that the values of f(x) approach a specific constant value l. This concept is crucial in understanding the behavior of functions at extreme values and helps determine whether a function approaches a finite number or diverges as x grows larger. Understanding this limit assists in analyzing horizontal asymptotes and overall function behavior for large inputs.
Limit at Infinity: A limit at infinity refers to the behavior of a function as its input approaches infinity or negative infinity. It helps to understand how functions behave for extremely large or small values and can indicate horizontal asymptotes, which are important in graphing functions and analyzing their long-term behavior.
Limit Graph: A limit graph visually represents the behavior of a function as it approaches a certain value or infinity. It helps to illustrate concepts such as infinite limits, which occur when the function approaches an unbounded value, and limits at infinity, where the input values grow without bound. By observing the limit graph, one can determine the behavior of a function around vertical and horizontal asymptotes, revealing essential information about continuity and discontinuity.
Limit Law: Limit laws are fundamental rules that describe how limits behave and can be manipulated in calculus. These laws provide a systematic approach to evaluating limits, allowing us to simplify expressions and analyze the behavior of functions as they approach specific points or infinity. By using limit laws, one can determine the limit of a function more easily by breaking it down into simpler components or combining them appropriately.
Oblique Asymptote: An oblique asymptote, also known as a slant asymptote, occurs when the graph of a rational function approaches a straight line as the input values approach positive or negative infinity. This happens specifically when the degree of the numerator is exactly one greater than the degree of the denominator, leading to a linear approximation of the function's behavior at extreme values. Identifying oblique asymptotes is essential for understanding the long-term behavior of functions and interpreting their graphs accurately.
One-Sided Limits: One-sided limits refer to the value that a function approaches as the input approaches a particular point from either the left or the right. This concept is crucial in understanding how functions behave near points of discontinuity and helps in analyzing limits involving infinite behavior or indeterminate forms, providing a clearer picture of a function's behavior in these contexts.
Rational Functions: Rational functions are functions that can be expressed as the ratio of two polynomials, typically written in the form $$R(x) = \frac{P(x)}{Q(x)}$$ where $$P(x)$$ and $$Q(x)$$ are polynomials and $$Q(x) \neq 0$$. These functions are important because they can exhibit unique behaviors such as asymptotes, discontinuities, and varying end behavior. Understanding rational functions helps in analyzing their limits, differentiating them, and applying the Intermediate Value Theorem effectively.
Squeeze Theorem: The Squeeze Theorem states that if a function is 'squeezed' between two other functions that converge to the same limit at a certain point, then the squeezed function must also converge to that limit at that point. This concept helps in evaluating limits, especially when direct substitution fails or the behavior of the function is difficult to determine.
Two-sided limits: A two-sided limit is a concept in calculus that refers to the value that a function approaches as the input approaches a certain point from both the left and the right sides. This means that for a limit to exist at a particular point, both the left-hand limit and the right-hand limit must be equal. Two-sided limits play a crucial role in analyzing functions, especially when determining continuity and dealing with infinite limits and limits at infinity.
Vertical Asymptote: A vertical asymptote is a vertical line $x = a$ where a function approaches infinity or negative infinity as the input approaches the value 'a'. This behavior occurs when the function is undefined at that point, usually due to a zero in the denominator of a rational function. Vertical asymptotes help in understanding the limits of functions and analyzing their behavior as they approach certain x-values.
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