Source: MTH 132, Chapter 1 (Sections 5.1–5.3) | Michigan State University
Tags: limit, left-hand limit, right-hand limit, one-sided limit, two-sided limit, infinite limit, vertical asymptote, DNE, does not exist, calculus 1, MTH 132, MSU
Difficulty: Introductory to Intermediate. Prerequisites: understanding of function notation, piecewise functions, and the tangent/secant material from Sections 4.1–4.3.
Limits are the foundation of every major concept in calculus: derivatives, integrals, and series all rest on them. This section introduces the intuitive idea of a limit (what value does f(x) approach as x gets close to some number a?), then formalises it with left-hand and right-hand limits. It also covers infinite limits and vertical asymptotes, which describe what happens when function values blow up. If you skipped the tangent line material, go back: the tangent line's slope is defined as a limit, so this section explains the machinery behind that idea.
A limit describes the value a function approaches as the input gets close to a target, regardless of whether the function is defined there. One-sided limits look at approach from one direction only, and the two-sided limit exists only when both one-sided limits agree. When a function's values grow without bound near a point, the limit is infinite and the graph has a vertical asymptote there.
Limit (two-sided)
We write lim(x→a) f(x) = L if we can force the values of f(x) as close to L as we like by requiring x to be sufficiently close (but not equal) to a.
In simple terms, as x sneaks up on a from either side, f(x) settles toward L.
Left-hand limit
Written lim(x→a⁻) f(x) = L. The values of f(x) approach L as x approaches a from the left (x < a).
Right-hand limit
Written lim(x→a⁺) f(x) = L. The values of f(x) approach L as x approaches a from the right (x > a).
Theorem connecting one-sided and two-sided limits
lim(x→a) f(x) = L if and only if lim(x→a⁻) f(x) = L and lim(x→a⁺) f(x) = L. Both one-sided limits must exist and be equal for the two-sided limit to exist.
"Does not exist" (DNE)
A limit does not exist when: (a) the left-hand and right-hand limits are different finite values (a jump), or (b) the function oscillates without settling, or (c) the function goes to +∞ or –∞ (infinite, not a real number).
Infinite limit
We write lim(x→a) f(x) = ∞ (or –∞) if f(x) can be made arbitrarily large (or arbitrarily negative) by taking x sufficiently close to a.
In simple terms, the function blows up near a, shooting toward positive or negative infinity.
Vertical asymptote
The line x = a is a vertical asymptote of y = f(x) if at least one of the one-sided limits at a is +∞ or –∞.
In simple terms, the graph shoots up or down without bound and never crosses the vertical line x = a (or at least approaches it explosively from one side).
To find lim(x→a⁻) f(x), trace the curve from the left toward x = a and read the y-value the curve approaches.
To find lim(x→a⁺) f(x), trace from the right.
The two-sided limit exists only when both one-sided limits match. A filled dot on the graph shows f(a), the actual function value, which may differ from the limit.
An open circle indicates the function is not defined at that point, or is defined at a different value.
For a piecewise function like f(x) = { x² + 4 if x < 1; 0 if x = 1; 3 – x if x > 1 }:
lim(x→1⁻) f(x): use the x < 1 piece, plug in x = 1 to get 1² + 4 = 5.
lim(x→1⁺) f(x): use the x > 1 piece, plug in x = 1 to get 3 – 1 = 2.
lim(x→1) f(x): the one-sided limits (5 and 2) disagree, so the two-sided limit does not exist.
A four-piece version adds a case for x > 2 (e.g. x/2). Each boundary point must be checked separately.
For functions of the form g(x)/h(x) where g(a) ≠ 0 and h(a) = 0, there is often a vertical asymptote at x = a.
To confirm, compute the left-hand and right-hand limits as x → a. Check the sign of the numerator and denominator on each side.
Example: for y = (x² + 2x + 1)/(x + x²) = (x + 1)²/[x(x + 1)] = (x + 1)/x (when x ≠ –1), the vertical asymptote is at x = 0. At x = –1, the factor cancels, so there is a hole (removable discontinuity), not a vertical asymptote.
Example: lim(x→1⁺) 5/(1 – x). As x → 1⁺, the denominator (1 – x) is small and negative, so the fraction is large and negative: the limit is –∞.
Vertical asymptote at x = a means at least one of: lim(x→a⁺) f(x) = ±∞ or lim(x→a⁻) f(x) = ±∞.
You can have the function going to +∞ from one side and –∞ from the other. The two-sided limit does not exist in either case, but the vertical asymptote is still there.
No new computation formulas are introduced in this section. The central results are:
lim(x→a) f(x) = L ⟺ lim(x→a⁻) f(x) = L and lim(x→a⁺) f(x) = L.
For g(x)/h(x) with g(a) ≠ 0 and h(a) = 0, expect a vertical asymptote at x = a. Verify by computing one-sided limits and checking signs.
Limits describe behaviour at boundaries: the temperature a heated rod approaches as time goes on, the concentration of a drug as it is metabolised toward zero, or the cost per unit as production volume grows large. Vertical asymptotes arise in physics whenever a denominator represents a distance that can shrink to zero, such as the gravitational force between two objects as the separation approaches zero, or the electric field strength near a point charge.
Students often think the limit of f(x) as x → a must equal f(a). It does not. The limit depends only on what happens near a, not at a. A function can have a limit of 5 at x = 2 even if f(2) = 0 or f(2) is undefined.
Students sometimes write lim(x→a) f(x) = DNE and treat DNE as a number. "Does not exist" is a conclusion, not a value. Never plug DNE into arithmetic.
A vertical asymptote does not mean the limit equals infinity in the same way a limit equals 3. Writing lim = ∞ is a shorthand saying the function grows without bound; the limit, strictly speaking, does not exist as a finite number.
When checking a piecewise function at a boundary, students sometimes plug the boundary value into the wrong piece. Always match the direction of approach (left or right) to the correct branch of the piecewise definition.
⚠️ Reading limits from a graph is almost guaranteed on MTH 132 exams. Practise distinguishing open dots (limit point) from filled dots (function value).
⚠️ Piecewise limit problems appear regularly. The exam will test whether both one-sided limits agree.
⚠️ You will be asked to identify vertical asymptotes and justify with one-sided limits showing ±∞.
⚠️ A question may give you f(a) and lim(x→a) f(x) separately and ask whether they are equal. The answer can be no.
True or false: lim(x→a) f(x) = f(a) always.
False. The limit and the function value can differ.
Fill in the blank: lim(x→a) f(x) exists if and only if the ____ limit and the ____ limit both exist and are ____.
Left-hand, right-hand, equal.
True or false: if lim(x→3⁺) f(x) = ∞, then x = 3 is a vertical asymptote.
True.
Fill in the blank: for g(x)/h(x), if g(a) ≠ 0 and h(a) = 0, there is likely a ____ at x = a.
Vertical asymptote.
True or false: lim(x→a) f(x) = ∞ means the limit exists and equals infinity.
False. Writing ∞ is shorthand for unbounded growth; the limit does not exist as a finite number.
Q: For f(x) = { x² + 4 if x < 1; 0 if x = 1; 3 – x if x > 1 }, find lim(x→1) f(x).
A: lim(x→1⁻) f(x) = 1 + 4 = 5. lim(x→1⁺) f(x) = 3 – 1 = 2. Since 5 ≠ 2, the two-sided limit does not exist.
Q: Evaluate lim(x→1⁺) 5/(1 – x).
A: As x → 1⁺, the denominator (1 – x) is small and negative. The numerator is 5 (positive). A positive number divided by a small negative number gives a large negative number, so the limit is –∞.
Q: Find the vertical asymptotes of y = (x² + 2x + 1)/(x + x²).
A: Factor: numerator = (x + 1)², denominator = x(x + 1). Cancel the common (x + 1) for x ≠ –1, leaving (x + 1)/x. The denominator is zero at x = 0, and the numerator is 1 ≠ 0 there, so x = 0 is a vertical asymptote. At x = –1, the common factor cancels, so there is a removable discontinuity (a hole), not a vertical asymptote.
Q: Evaluate lim(x→0⁻) (x² + 2x + 1)/(x + x²).
A: After cancelling, the simplified form is (x + 1)/x. As x → 0⁻, numerator → 1 (positive) and denominator → 0⁻ (small negative). So the limit is –∞.
Q: A graph of f shows an open circle at (2, 1) and a filled dot at (2, 3). What is lim(x→2) f(x) and what is f(2)?
A: lim(x→2) f(x) = 1 (where the curve approaches). f(2) = 3 (the filled dot).
This connects directly to Section 6 (Calculating Limits Using the Limit Laws), which gives you the algebraic tools to evaluate limits without a graph. It also connects forward to the definition of continuity in Section 8: a function is continuous at a precisely when lim(x→a) f(x) = f(a). Vertical asymptotes reappear when you study rational functions in detail and again when sketching curves with derivatives in Chapter 3.
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