Glossary

2.2 The Limit of a Function

4 min read•june 24, 2024

Limits are the foundation of calculus, describing how functions behave as they approach specific values. They help us understand behavior near critical points, even when the function isn't defined there.

Limits have real-world applications in physics, engineering, and economics. They're used to analyze velocities, optimize designs, and predict market trends. Understanding limits is crucial for grasping more advanced calculus concepts.

Understanding Limits

Concept and notation of limits

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  • represents the value a function approaches as the input (xx) gets arbitrarily close to a specific value without necessarily reaching it
    • Describes function behavior near a point rather than at the exact point
  • Limit notation for a function f(x)f(x) as xx approaches aa:
    • lim⁡x→af(x)=L\lim_{x \to a} f(x) = L
    • As xx gets closer to aa, f(x)f(x) gets closer to LL (the "target value")
  • Limits provide insight into function behavior even if the function never reaches the limit value

Limit estimation techniques

  • Estimate limits using tables of values:
    • Create a table with xx values approaching the point of interest from both sides and corresponding f(x)f(x) values
    • If f(x)f(x) values approach a single value as xx gets closer, that value is the estimated limit
  • Estimate limits using graphs:
    • Visually observe the graph's behavior near the point of interest
    • If the graph approaches a specific value from both sides, that value is the estimated limit
    • Graphs with holes, jumps, or asymptotes at the point of interest may indicate non-existent or different limits

Types of Limits and Their Properties

One-sided vs two-sided limits

  • Two-sided limit: function input approaches a point from both positive and negative directions
    • Exists only if one-sided limits are equal
  • : function input approaches a point from one direction (left or right)
    • Left-hand limit: lim⁡x→a−f(x)\lim_{x \to a^-} f(x), xx approaches aa from values less than aa
    • Right-hand limit: lim⁡x→a+f(x)\lim_{x \to a^+} f(x), xx approaches aa from values greater than aa

Non-existent limits

  • Limits may not exist when:
    • Function has a at the point of interest
    • Function has a jump discontinuity at the point of interest
    • Function oscillates or behaves erratically near the point of interest
    • One-sided limits are unequal (left-hand and right-hand limits differ)

Infinite limits and vertical asymptotes

  • Infinite limit: function grows without bound (positively or negatively) as input approaches a specific value
    • lim⁡x→af(x)=∞\lim_{x \to a} f(x) = \infty: function increases without bound as xx approaches aa
    • lim⁡x→af(x)=−∞\lim_{x \to a} f(x) = -\infty: function decreases without bound as xx approaches aa
  • Vertical asymptotes: lines that a function approaches but never touches as input approaches a specific value
    • If lim⁡x→a−f(x)=±∞\lim_{x \to a^-} f(x) = \pm\infty or lim⁡x→a+f(x)=±∞\lim_{x \to a^+} f(x) = \pm\infty, function has a at x=ax = a

Applications of Limits

Real-world applications of limits

  • Analyze physical systems:
    • Velocity or acceleration at an instant by taking the limit of average velocity or acceleration as time interval approaches zero
    • Maximum load a bridge can support by taking the limit of deflection as load approaches a critical value
    • Optimal container dimensions to maximize volume and minimize surface area by taking the limit of volume-to-surface-area ratio as dimensions approach specific values

Function behavior near critical points

  • Critical points: xx values where function is not differentiable or derivative is zero
  • Limits analyze function behavior near critical points:
    • If lim⁡x→a−f(x)=lim⁡x→a+f(x)=f(a)\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a), function is continuous at x=ax = a (no gaps, jumps, or asymptotes)
    • If one-sided limits exist but are unequal, function has a jump discontinuity at the critical point
    • If one or both one-sided limits are infinite, function has a vertical asymptote at the critical point

Function Properties and Limits

Continuity and differentiability

  • : a function is continuous at a point if the limit exists and equals the function value at that point
    • Limits help determine if a function is continuous by comparing the limit to the function value
  • Differentiability: a function is differentiable at a point if the limit of the difference quotient exists at that point
    • Limits are crucial in defining derivatives and determining where a function is differentiable

Domain, range, and asymptotes

  • : the set of all possible input values (x-values) for a function
    • Limits can help identify points where the function is undefined or approaches infinity
  • : the set of all possible output values (y-values) for a function
    • Horizontal asymptotes, found using limits, can indicate bounds on the range
  • Asymptotes: lines that a function approaches but never reaches
    • Vertical asymptotes: found using limits as x approaches a specific value
    • Horizontal asymptotes: found using limits as x approaches positive or negative infinity

Key Terms to Review (15)

Absolute value function: An absolute value function is a function that contains an algebraic expression within absolute value symbols. The output of the absolute value function is always non-negative.
Constant multiple law for limits: The Constant Multiple Law for limits states that the limit of a constant multiplied by a function is equal to the constant multiplied by the limit of the function. Mathematically, if $\lim_{{x \to c}} f(x) = L$, then $\lim_{{x \to c}} [k \cdot f(x)] = k \cdot L$ where $k$ is a constant.
Continuity: Continuity is a fundamental concept in calculus that describes the smoothness and uninterrupted nature of a function. It is a crucial property that allows for the application of calculus techniques and the study of limits, derivatives, and integrals.
Continuity over an interval: Continuity over an interval means that a function is continuous at every point within a given interval. This implies that the function has no breaks, jumps, or holes in that interval.
Domain: The domain of a function is the set of all possible input values (typically $x$-values) for which the function is defined. It represents all the values that can be plugged into the function without causing any undefined behavior.
Domain: The domain of a function refers to the set of all possible input values for which the function is defined. It represents the range of values that the independent variable can take on, and it is a crucial concept in understanding the behavior and properties of functions.
Factoring: Factoring is the process of breaking down a mathematical expression, such as a polynomial or an algebraic expression, into a product of simpler factors. This technique is essential in various mathematical contexts, including the analysis of functions, limits, and asymptotes.
Function: A function is a mathematical relationship between two or more variables, where one variable (the dependent variable) depends on the value of the other variable(s) (the independent variable(s)). Functions are central to the study of calculus, as they provide the foundation for understanding concepts like limits, derivatives, and integrals.
Infinite limits: An infinite limit exists when the value of a function increases or decreases without bound as the input approaches a certain point. This can be represented mathematically by $\lim_{{x \to a}} f(x) = \infty$ or $\lim_{{x \to a}} f(x) = -\infty$.
Intuitive definition of the limit: The intuitive definition of the limit describes the value that a function approaches as the input approaches a certain point. It helps us understand the behavior of functions near specific points.
Limit: In mathematics, the limit of a function is a fundamental concept that describes the behavior of a function as its input approaches a particular value. It is a crucial notion that underpins the foundations of calculus and serves as a building block for understanding more advanced topics in the field.
One-sided limit: A one-sided limit is the value that a function approaches as the input approaches a given value from one side—either from the left or the right. It is denoted as $\lim_{{x \to c^-}} f(x)$ for the left-hand limit and $\lim_{{x \to c^+}} f(x)$ for the right-hand limit.
Range: Range refers to the set of all possible output values (or 'y' values) that a function can produce based on its domain (the set of input values). Understanding the range helps us grasp how a function behaves, what outputs are attainable, and the limitations on those outputs.
Vertical asymptote: A vertical asymptote is a line $x = a$ where the function $f(x)$ approaches positive or negative infinity as $x$ approaches $a$. Vertical asymptotes occur at values of $x$ that make the denominator of a rational function zero, provided that the numerator does not also become zero at those points.
Vertical Asymptote: A vertical asymptote is a vertical line that a function's graph approaches but never touches. It represents the value of the independent variable where the function becomes undefined or experiences a vertical discontinuity.
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