Epsilon-Delta Variations: One-Sided and Infinite Limits

5 min read

Beyond the Standard Definition

The standard epsilon-delta definition handles $\lim_{x \to a} f(x) = L$ where $x$ approaches $a$ from both sides and $L$ is a finite number. But what about:

• One-sided limits: $x$ approaches only from the left ($x \to a^-$) or right ($x \to a^+$)
• Infinite limits: The limit “equals” $\infty$ or $-\infty$
• Limits at infinity: $x \to \infty$ or $x \to -\infty$

Each variation follows the same logical structure. Just change what we’re controlling and what we’re guaranteeing.

Prerequisite Map

Quick Reference

Key Concepts

Comparison of All Definitions

One-Sided Limits

Left-hand limit: $\lim_{x \to a^-} f(x) = L$

$$\forall \varepsilon > 0,\ \exists \delta > 0 \text{ such that } a - \delta < x < a \Rightarrow \vert f(x) - L\vert < \varepsilon$$

        f(x)
          │
    L+ε ──┼────────────────
          │    ●●●●●●
      L ──┼───●───────────○  (open at x=a)
          │  ●
    L-ε ──┼─●─────────────
          │
          └───────────┼───→ x
                a-δ   a
              ◄───────►
              only x < a

Right-hand limit: $\lim_{x \to a^+} f(x) = L$

$$\forall \varepsilon > 0,\ \exists \delta > 0 \text{ such that } a < x < a + \delta \Rightarrow \vert f(x) - L\vert < \varepsilon$$

        f(x)
          │
    L+ε ──┼────────────────
          │         ●●●●●●
      L ──┼───○────●───────  (open at x=a)
          │         ●
    L-ε ──┼──────────●─────
          │
          └─────┼───────┼──→ x
                a     a+δ
                ◄───────►
                only x > a

Key difference: Replace “$0 < \vert x - a\vert < \delta$” with:

• Left: “$a - \delta < x < a$” (x is less than a)
• Right: “$a < x < a + \delta$” (x is greater than a)

Infinite Limits

Definition: $\lim_{x \to a} f(x) = \infty$

$$\forall M > 0,\ \exists \delta > 0 \text{ such that } 0 < \vert x - a\vert < \delta \Rightarrow f(x) > M$$

Key changes:

• Replace $\varepsilon$ (small tolerance) with $M$ (large threshold)
• Replace “$\vert f(x) - L\vert < \varepsilon$” with “$f(x) > M$”
• The “game” is now: no matter how large an $M$ you demand, I can make $f(x)$ exceed it by getting $x$ close enough to $a$

    f(x)
      │             │
      │             │
    M ├─────────────┼─────
      │           ╱ │ ╲
      │         ╱   │   ╲
      │       ╱     │     ╲
      │     ╱       │       ╲
      └─────────────┼─────────→ x
                    a
                 ◄──┴──►
                   δ

For $\lim_{x \to a} f(x) = -\infty$: Change “$f(x) > M$” to “$f(x) < -M$”.

Limits at Infinity

Definition: $\lim_{x \to \infty} f(x) = L$

$$\forall \varepsilon > 0,\ \exists N > 0 \text{ such that } x > N \Rightarrow \vert f(x) - L\vert < \varepsilon$$

Key changes:

• Replace $\delta$ with $N$ (a large threshold for $x$)
• Replace “$0 < \vert x - a\vert < \delta$” with “$x > N$”
• We’re saying: for any $\varepsilon$, making $x$ large enough keeps $f(x)$ within $\varepsilon$ of $L$

    f(x)
      │
  L+ε ├───────────────●●●●●●●●●
      │            ●●●
    L ├──────────●●────────────
      │        ●●
  L-ε ├──────●●────────────────
      │    ●●
      │  ●●
      └─────┼─────────────────→ x
            N
            ◄─────────────────►
                  x > N

Worked Example: One-Sided Limit

Prove: $\lim_{x \to 0^+} \sqrt{x} = 0$

Proof: Let $\varepsilon > 0$ be given. Choose $\delta = \varepsilon^2$.

Suppose $0 < x < \delta = \varepsilon^2$.

Then $\sqrt{x} < \sqrt{\varepsilon^2} = \varepsilon$ (since $\sqrt{}$ is increasing).

So $\vert \sqrt{x} - 0\vert = \sqrt{x} < \varepsilon$.

Therefore $\lim_{x \to 0^+} \sqrt{x} = 0$. $\square$

Worked Example: Infinite Limit

Prove: $\lim_{x \to 0} \frac{1}{x^2} = \infty$

Proof: Let $M > 0$ be given. Choose $\delta = \frac{1}{\sqrt{M}}$.

Suppose $0 < \vert x\vert < \delta = \frac{1}{\sqrt{M}}$.

Then $\vert x\vert ^2 < \frac{1}{M}$, so $\frac{1}{x^2} = \frac{1}{\vert x\vert ^2} > M$.

Therefore $\lim_{x \to 0} \frac{1}{x^2} = \infty$. $\square$

Practice Problems

Conceptual Check (CCI-Style)

Question: For which of the following can we write a formal epsilon-delta style definition?

1. $\lim_{x \to 5} f(x) = 3$
2. $\lim_{x \to 5} f(x) = \infty$
3. $\lim_{x \to \infty} f(x) = 3$
4. $\lim_{x \to \infty} f(x) = \infty$
5. All of the above

Summary Table

Common Misconceptions

Mastery Checklist

• I can state the definition for left-hand limits ($x \to a^-$)
• I can state the definition for right-hand limits ($x \to a^+$)
• I can state the definition for infinite limits ($f(x) \to \infty$)
• I can state the definition for limits at infinity ($x \to \infty$)
• I can prove one-sided limits using the modified definition
• I can prove infinite limits by finding $\delta$ in terms of $M$
• I can prove that two-sided limits equal both one-sided limits

Mental Model

The Universal Structure:

Every epsilon-delta variation has the same logical skeleton:

“For every challenge (small $\varepsilon$ or large $M$), there exists a response ($\delta$ or $N$) such that satisfying the input condition guarantees the output condition.”

Connections

Looking back:

• Epsilon-Delta Definition is the foundation that these variations modify.
• One-Sided Limits gave the intuition; now we have precision.

Looking ahead:

• These definitions prove that vertical asymptotes occur where $\lim_{x \to a} f(x) = \pm\infty$.
• Horizontal asymptotes are characterized by $\lim_{x \to \pm\infty} f(x) = L$.

Last updated: 2026-01-22