Calculus

The Limit of a Function: An Intuitive Journey

Imron Rosyadi

The Limit of a Function: An Intuitive Journey

Understanding Calculus Foundations

Today’s Learning Objectives

By the end of this session, you will be able to:

• Describe the limit of a function using correct notation.
• Estimate limits through tables of values and graphs.
• Define one-sided limits and illustrate their relationship to two-sided limits.
• Explain infinite limits and identify vertical asymptotes.
• Apply these concepts to real-world scenarios, including physics.

These objectives set the stage for our exploration of limits, starting intuitively and moving towards more formal concepts. They align directly with the learning goals from the OpenStax Calculus text.

What is a Limit?

The concept of a limit is fundamental to calculus. It describes the behavior of a function as the input approaches a particular value. Crucially, it’s about what the function is approaching, not necessarily what it is at that exact point.

Let’s consider three functions near \(x=2\): \[ f(x)=\frac{x^2-4}{x-2}, \quad g(x)=\frac{|x-2|}{x-2}, \quad \text{and} \quad h(x)=\frac{1}{(x-2)^2} \]

Note

Notice that all these functions are undefined at \(x=2\). This is precisely where limits become valuable!

Visualizing Limit Behavior

Let’s look at the graphs for our example functions.

Three graphs of functions illustrating different limit behaviors near x=2.

Function f(x)

As x approaches 2, f(x) approaches 4.

This is a removable discontinuity.

\[ \lim_{x \to 2} f(x) = 4 \]

Function g(x)

As x approaches 2 from the left, g(x) approaches -1.

As x approaches 2 from the right, g(x) approaches 1.

No single value is approached. \[ \lim_{x \to 2} g(x) \text{ DNE} \]

Visualizing Limit Behavior (Cont.)

The third function:

Graph of h(x) = 1/(x-2)^2 showing an infinite limit at x=2.

Function h(x)

As x approaches 2 from both sides, h(x) values become infinitely large.

This indicates an infinite limit. \[ \lim_{x \to 2} h(x) = +\infty \]

Important

Even though the function values “approach” infinity, we say the limit “does not exist” in the sense of being a finite number but describe its behavior as +∞.

Intuitive & Formal Definition of a Limit

Intuitive Understanding:

The limit of a function \(f(x)\) at a number \(a\) is the single real number \(L\) that the functional values approach as the \(x\)-values approach \(a\), provided such a real number \(L\) exists.

Formal Notation:

If \(f(x)\) approaches \(L\) as \(x\) approaches \(a\), we write: \[ \lim_{x \to a} f(x) = L \]

The formal definition (epsilon-delta) provides mathematical rigor but can be complex for initial understanding. For now, we focus on the intuitive idea. We’ll revisit the formal definition later in the course.

Evaluating Limits Using Graphs

Sometimes, the algebraic expression for a function is not known, but its graph is. We can infer limits directly from the graphical behavior.

Consider the graph of \(g(x)\):

At \(x=-1\):

• As \(x \to -1\) from both sides, the function values approach 3.
• However, \(g(-1) = 4\).

Therefore, \(\lim_{x \to -1} g(x) = 3\).

Important

The value of the function at the point (\(f(a)\)) does not always equal the limit as x approaches the point (\(\lim_{x \to a} f(x)\)).

Two Important Limits

These two limits are foundational:

1. Limit of \(x\): For any real number \(a\), as \(x\) approaches \(a\), the value of \(x\) approaches \(a\). \[ \lim_{x \to a} x = a \]

2. Limit of a Constant: For any real number \(a\) and any constant \(c\), as \(x\) approaches \(a\), the value of \(c\) remains \(c\). \[ \lim_{x \to a} c = c \]

These simple limits form the building blocks for more complex limit computations later on. The graph for \(f(x)=x\) is a straight line, and for \(f(x)=c\) it is a horizontal line. The limit is intuitively clear.

The Existence of a Limit

For a limit to exist at a point, the functional values must approach a single real-number value at that point. If they do not, the limit does not exist (DNE).

The Existence of a Limit: Visualization

Here’s a plot of \(\sin(1/x)\) to visually reinforce the non-existence of the limit.

One-Sided Limits

Sometimes, a limit fails to exist, but we can still describe the function’s behavior from specific directions.

• Limit from the Left: \(f(x)\) approaches \(L\) as \(x\) approaches \(a\) from values less than \(a\) (\(x < a\)). \[ \lim_{x \to a^-} f(x) = L \]

• Limit from the Right: \(f(x)\) approaches \(L\) as \(x\) approaches \(a\) from values greater than \(a\) (\(x > a\)). \[ \lim_{x \to a^+} f(x) = L \]

Recall the function \(g(x) = |x-2|/(x-2)\) from the beginning. Its graph showed a clear difference in values when approaching from the left vs. the right.

Example: One-Sided Limits with a Piecewise Function

Consider the piecewise function: \[ f(x)= \begin{cases} x+1 & \text{if } x < 2 \\ x^2-4 & \text{if } x \ge 2 \end{cases} \] Let’s find \(\lim_{x \to 2^-} f(x)\) and \(\lim_{x \to 2^+} f(x)\).

From the left, \(f(x)\) approaches 3. From the right, \(f(x)\) approaches 0. Since these are different, the two-sided limit \(\lim_{x \to 2} f(x)\) DNE.

Plotting the Piecewise Function

The discontinuity is clear in the graph.

Graph of the piecewise function f(x).

Note

Notice the open circle at \((2,3)\) and the closed circle at \((2,0)\). This signifies the values taken by the function at the boundary.

Relationship Between One-Sided and Two-Sided Limits

The existence of a two-sided limit is directly tied to its one-sided counterparts.

\[ \lim_{x \to a} f(x) = L \quad \text{if and only if} \quad \lim_{x \to a^-} f(x) = L \text{ and } \lim_{x \to a^+} f(x) = L \]

This means:

• If the left-hand limit equals the right-hand limit, the two-sided limit exists and is equal to that common value.
• If the left-hand limit and right-hand limit are different, or if either one does not exist, then the two-sided limit does not exist.

Infinite Limits

Limits can also “approach” infinity, indicating that the function’s values grow without bound (positive infinity) or decrease without bound (negative infinity).

Example: \(\lim_{x \to 2} \frac{1}{(x-2)^2}\)

As \(x\) approaches 2, \((x-2)^2\) approaches 0 (always positive), so \(\frac{1}{(x-2)^2}\) becomes very large. \[ \lim_{x \to 2} \frac{1}{(x-2)^2} = +\infty \]

Formal Definitions of Infinite Limits:

• \(\lim_{x \to a^-} f(x) = +\infty\) or \(\lim_{x \to a^+} f(x) = +\infty\) means \(f(x)\) increases without bound.
• \(\lim_{x \to a^-} f(x) = -\infty\) or \(\lim_{x \to a^+} f(x) = -\infty\) means \(f(x)\) decreases without bound.
• \(\lim_{x \to a} f(x) = +\infty\) or \(\lim_{x \to a} f(x) = -\infty\) means both one-sided limits go to the same infinity.

Example: Recognizing an Infinite Limit for \(1/x\)

Let’s evaluate \(\lim_{x \to 0} \frac{1}{x}\).

From the left, \(f(x) \to -\infty\). From the right, \(f(x) \to +\infty\). Since they go to different infinities, \(\lim_{x \to 0} \frac{1}{x}\) DNE (in a general sense), though we describe the one-sided behaviors explicitly.

Visualization for \(1/x\)

Vertical Asymptotes

A vertical asymptote is a vertical line \(x=a\) that a function approaches arbitrarily closely but never touches, as the function’s value tends towards positive or negative infinity.

Precisely, the line \(x=a\) is a vertical asymptote of \(f(x)\) if any of the following conditions hold: \[ \lim_{x \to a^-} f(x) = \pm \infty \quad \text{or} \quad \lim_{x \to a^+} f(x) = \pm \infty \quad \text{or} \quad \lim_{x \to a} f(x) = \pm \infty \]

Caution

Remember: having an infinite limit means the two-sided limit does not exist as a finite value, but it describes a specific type of unbounded behavior.

Diagram: Infinite Limits & Vertical Asymptotes

graph TD
    A"[Function f(x) near point 'a'"] --> B{Does L exist?};
    B -- Yes, L is finite --> C["Limit exists: lim f(x) = L"];
    B -- No --> D{"Does f(x) go to infinity?"};
    D -- Yes, from both sides to +inf --> E["lim f(x) = +inf"];
    D -- Yes, from both sides to -inf --> F["lim f(x) = -inf"];
    D -- No, different infinities or oscillates --> G["lim f(x) DNE (oscillates or jump)"];
    E -- Implies --> H[Vertical Asymptote at x=a];
    F -- Implies --> H;
    I["Consider specifically 1/(x-a)^n"] --> J{n is even?};
    J -- Yes, n=2,4,... --> K["lim 1/(x-a)^n = +inf, VA x=a"];
    J -- No, n is odd --> L{Approach from left/right?};
    L -- From left (x<a) --> M["lim 1/(x-a)^n = -inf"];
    L -- From right (x>a) --> N["lim 1/(x-a)^n = +inf"];
    M -- Implies --> H;
    N -- Implies --> H;

Case Study: Einstein’s Equation and the Speed of Light

Albert Einstein’s theory of relativity states that an object’s mass \(m\) increases with its velocity \(v\): \[ m = \frac{m_0}{\sqrt{1 - \frac{v^2}{c^2}}} \] where \(m_0\) is the rest mass and \(c\) is the speed of light.

What happens as \(v\) approaches \(c\)? Let’s analyze the ratio of masses, \(m/m_0 = \frac{1}{\sqrt{1 - (v/c)^2}}\). We are interested in \(\lim_{v \to c^-} \frac{1}{\sqrt{1 - (v/c)^2}}\).

The equation suggests a physical limit. As speed approaches the speed of light, the mass approaches infinity. This result implies that it is impossible for any object with mass to reach or exceed the speed of light, as it would require infinite energy.

Interactive Visualization: Relativistic Mass Increase

Let’s see how the mass ratio behaves as \(v/c\) approaches 1.

Conclusion of Einstein’s Equation

The graph clearly shows a vertical asymptote at \(v/c=1\). This means: \[ \lim_{v \to c^-} \frac{m_0}{\sqrt{1 - \frac{v^2}{c^2}}} = +\infty \] As an object’s speed approaches the speed of light, its mass approaches infinity. Since infinite mass is physically impossible, no object can reach or exceed the speed of light.

Important

This is a powerful real-world application of infinite limits and vertical asymptotes! It defines a fundamental physical constant and a universal speed limit.

Summary & Key Takeaways

Core Concepts

• Limit: Value a function approaches as input approaches a point. \[ \lim_{x \to a} f(x) = L \]
• One-Sided Limits: Approaching from left (\(x \to a^-\)) or right (\(x \to a^+\)).
• Existence: Two-sided limit exists if and only if both one-sided limits exist and are equal.
• Infinite Limits: Function values grow/decrease without bound (\(\pm \infty\)).
• Vertical Asymptote: A vertical line (\(x=a\)) where a function has an infinite limit.

Methods & Tools

• Tables of Values: Numerical approximation of limits.
• Graphs: Visual estimation of limits and discontinuities.
• Pyodide: Interactive Python for numerical evaluations and plotting.
• Plotly: Creating dynamic and interactive visualizations.
• Mermaid: Diagramming conceptual flows.

Think & Discuss

Consider the function \(f(x) = \frac{x^2 - 1}{|x-1|}\).

1. What are the one-sided limits \(\lim_{x \to 1^-} f(x)\) and \(\lim_{x \to 1^+} f(x)\)?
2. Does \(\lim_{x \to 1} f(x)\) exist? Why or why not?
3. How would the graph look around \(x=1\)?

Encourage students to discuss or use pen/paper/calculator to explore this function. They should simplify \(|x-1|\) by considering \(x>1\) and \(x<1\). For \(x>1\), \(|x-1|=x-1\), so \(f(x) = (x^2-1)/(x-1) = (x-1)(x+1)/(x-1) = x+1\). For \(x<1\), \(|x-1|=-(x-1)\), so \(f(x) = (x^2-1)/-(x-1) = (x-1)(x+1)/-(x-1) = -(x+1)\). So, \(\lim_{x \to 1^+} f(x) = 1+1=2\). And \(\lim_{x \to 1^-} f(x) = -(1+1)=-2\). Since the one-sided limits are different, the two-sided limit does not exist. The graph would have a jump discontinuity.