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Calculus I Foundations (MATH 151) · Module 1 of 11

Limits: Foundations & Techniques

What a limit means, and the algebraic techniques for evaluating one when substitution fails.

In this module
  1. The Intuitive Idea of a Limit
  2. Evaluating Limits Algebraically
  3. Limits at Infinity & Horizontal Asymptotes
  4. Infinite Limits & Vertical Asymptotes
  5. Check yourself

The Intuitive Idea of a Limit

  • lim(x→a) f(x) = L means f(x) gets arbitrarily close to L as x gets arbitrarily close to a, regardless of whether f(a) itself is defined or equals L.
  • One-sided limits — from the left, lim(x→a⁻), and from the right, lim(x→a⁺) — must both equal L for the two-sided limit to exist.
A limit at a hole
xyhole: limit = 2
f(x) = (x² − 1)/(x − 1) is undefined at x = 1, but as x approaches 1 from either side f(x) approaches 2. So the limit is 2 even though f(1) doesn't exist.
Worked example

If lim(x→2⁻) f(x) = 3 and lim(x→2⁺) f(x) = 5, does lim(x→2) f(x) exist?

Solution. No — since the left-hand and right-hand limits disagree (3 ≠ 5), the two-sided limit does not exist at x = 2.

Evaluating Limits Algebraically

  • Direct substitution works when the function is continuous at that point; if substitution gives 0/0, an indeterminate form, try factoring and canceling, or rationalizing by multiplying by a conjugate.
Worked example

Evaluate lim(x→3) (x² − 9)/(x − 3).

Solution. Direct substitution gives 0/0. Factor: (x² − 9)/(x − 3) = (x − 3)(x + 3)/(x − 3) = x + 3 for x ≠ 3. The limit is 3 + 3 = 6.

Limits at Infinity & Horizontal Asymptotes

  • lim(x→∞) f(x) describes a function's end behavior; for rational functions, compare the degree of the numerator and denominator: if equal, the limit is the ratio of leading coefficients; if the denominator's degree is higher, the limit is 0.
Worked example

Find lim(x→∞) (3x² + 1)/(x² − 5).

Solution. The numerator and denominator have the same degree (2), so the limit is the ratio of leading coefficients: 3/1 = 3.

Infinite Limits & Vertical Asymptotes

  • An infinite limit, like lim(x→a) f(x) = ∞, describes a vertical asymptote at x = a, occurring where the denominator approaches 0 but the numerator does not.
Worked example

What does it mean if lim(x→2⁺) f(x) = ∞?

Solution. As x approaches 2 from the right, f(x) grows without bound, indicating a vertical asymptote at x = 2 that the function shoots up toward from the right side.

Check yourself

2 of this module's 5 quiz questions. Pick your answers, then check them in the app, with an explanation for every question.

1. What must be true for a two-sided limit to exist at a point?

2. What technique helps evaluate a limit that gives the indeterminate form 0/0 upon direct substitution?

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