Difficulty: Foundational | Prerequisites: Algebra, basic trigonometry, function notation.
This is the opening chapter of single-variable calculus at Michigan State University (MATH 132), covering Stewart's Calculus sections 1.4 through 1.8. Limits are the mechanism that makes calculus work: every later topic (derivatives, integrals, series) rests on them. If you skipped a few weeks, start here, because nothing that follows makes sense without a solid grip on what a limit is and how to evaluate one. You should already be comfortable with factoring polynomials, rationalising denominators, and reading graphs.
A limit describes the value a function approaches as its input gets close to some number, even if the function is undefined there. Calculus uses limits to define both derivatives (rates of change) and integrals (accumulated totals). A function is continuous at a point when its limit there equals its actual value, and continuous functions obey the Intermediate Value Theorem, which guarantees they hit every value between their endpoints.
Limit
The value L that f(x) approaches as x gets arbitrarily close to a, written lim(x->a) f(x) = L. The function need not be defined at a itself. Think of it as the number the outputs are zeroing in on, even if the function never actually lands there.
One-sided limit
The value f(x) approaches when x comes from only one direction. lim(x->a+) means x approaches a from the right (values greater than a); lim(x->a-) means from the left. In simple terms, you are only looking at what happens on one side of the point.
Two-sided limit
The ordinary limit lim(x->a) f(x) = L, which exists only when the left and right one-sided limits both exist and are equal. If the two sides disagree, the two-sided limit does not exist.
Infinite limit
lim(x->a) f(x) = infinity means the outputs grow without bound as x approaches a. This is not a number but a shorthand for "f(x) can be made as large as you like." Think of it as the function blowing up near that input.
Vertical asymptote
A vertical line x = a where the function heads toward positive or negative infinity. Graphically, the curve shoots up or down and never crosses that line.
Difference quotient
The expression [f(a+h) - f(a)] / h, representing the average rate of change of f over a small interval. In simple terms, it is the slope of the secant line between two nearby points on the graph.
Instantaneous velocity
The limit of average velocities over shorter and shorter time intervals: v = lim(h->0) [f(a+h) - f(a)] / h. This is the speed at a single instant, not over a span of time.
Tangent line
The line that touches the curve y = f(x) at exactly one local point, with slope equal to the derivative. Its equation is y = f(a) + m(x - a), where m is the limit of secant slopes.
Continuous function
A function f(x) is continuous at x = a when lim(x->a) f(x) = f(a). All three conditions must hold: f(a) is defined, the limit exists, and they match. Think of it as being able to draw the graph through that point without lifting your pen.
Removable discontinuity
A hole in the graph where the limit exists but f(a) is either undefined or has the wrong value. The gap can be "removed" by redefining f(a) to equal the limit.
Jump discontinuity
A break where the left and right limits exist but are unequal. The graph has a visible step at that point.
Squeeze Theorem (Sandwich Theorem)
If f(x) <= g(x) <= h(x) near a, and lim(x->a) f(x) = lim(x->a) h(x) = L, then lim(x->a) g(x) = L. Think of it as trapping the mystery function between two known ones that both converge to the same value.
Epsilon-delta definition
The formal definition: lim(x->a) f(x) = L means for every error tolerance epsilon > 0, there exists an input accuracy delta > 0 such that 0 < |x - a| < delta forces |f(x) - L| < epsilon. In plain terms, you can make the output as close as you want to L by keeping the input close enough to a.
Intermediate Value Theorem (IVT)
If f is continuous on [a, b] and r is any value between f(a) and f(b), then there exists some c in (a, b) with f(c) = r. In simple terms, a continuous function that starts below a line and ends above it must cross that line somewhere in between.
The velocity problem motivates limits: if a particle is at position f(t) at time t, its average velocity from t = a to t = b is [f(b) - f(a)] / (b - a)
Instantaneous velocity is the limit of average velocities as the interval shrinks: v = lim(h->0) [f(a+h) - f(a)] / h
Example: a stone dropped from a bridge has f(t) = 16t^2. Average velocity from t = 3 to t = 3.1 is 97.6 ft/s. As h shrinks (0.1, 0.01, 0.001), the averages approach 96 ft/s exactly
Algebraically: [f(3+h) - f(3)] / h = 16[(3+h)^2 - 9] / h = 16(6+h) = 96 + 16h, which approaches 96 as h approaches 0
The tangent slope at (a, f(a)) equals the same limit, since slope = rise/run = [f(a+h) - f(a)] / h. The tangent line equation is y = f(a) + m(x - a)
lim(x->a) f(x) = L means f(x) can be forced as close as desired to L by making x sufficiently close to (but not equal to) a
The limit does not depend on the value of f(a), or on whether f(a) is defined at all
If f(x) fails to approach any single value near x = a, the limit does not exist
Example: sgn(x) = x/|x| oscillates between -1 and +1 near x = 0, so lim(x->0) sgn(x) does not exist
One-sided limits: lim(x->a+) f(x) = L means x approaches a from the right only. The two-sided limit exists exactly when left and right limits agree
Infinite limits: lim(x->a) f(x) = infinity means f(x) grows without bound near a. This produces a vertical asymptote at x = a
Example: lim(x->0) 1/|x| = infinity, but lim(x->0) 1/x does not equal infinity because the function goes to +infinity from the right and -infinity from the left
Vertical asymptotes arise where the denominator vanishes but the numerator does not. Track signs on each side to determine whether the function goes to +infinity or -infinity
Sum: lim [f(x) + g(x)] = lim f(x) + lim g(x)
Difference: lim [f(x) - g(x)] = lim f(x) - lim g(x)
Constant multiple: lim [c * f(x)] = c * lim f(x)
Product: lim [f(x) * g(x)] = lim f(x) * lim g(x)
Quotient: lim [f(x)/g(x)] = lim f(x) / lim g(x), provided the denominator limit is not zero
Power: lim [f(x)]^n = [lim f(x)]^n
Root: lim [f(x)]^(1/n) = [lim f(x)]^(1/n), provided the result is real
These laws mean most limits can be evaluated by direct substitution, unless substitution produces an indeterminate form like 0/0
When substitution gives 0/0, factor and cancel the vanishing factor before re-evaluating
Example: lim(x->1) (x^2 - 1)/(x - 1) = lim(x->1) (x-1)(x+1)/(x-1) = lim(x->1) (x+1) = 2
Rationalising trick: multiply by a conjugate radical to eliminate square roots
Example: lim(x->9) (x-9)/(sqrt(x)-3). Multiply top and bottom by (sqrt(x)+3) to get lim(x->9) (sqrt(x)+3) = 6
If f(x) <= g(x) <= h(x) near a, and both f and h approach L, then g must also approach L
Classic application: lim(x->0) x * sin(pi/x) = 0, because -|x| <= x * sin(pi/x) <= |x|, and both bounds go to 0
Note that lim(x->0) sin(pi/x) alone does not exist because the oscillations are unbounded in frequency. The extra factor of x tames the oscillation
The rigorous version: lim(x->a) f(x) = L means for every epsilon > 0, there exists delta > 0 such that whenever 0 < |x - a| < delta, we have |f(x) - L| < epsilon
This is an error-control problem: given any desired output accuracy (epsilon), find the required input accuracy (delta)
Example: prove lim(x->5) (3x - 2) = 13. We need |(3x-2) - 13| < epsilon, which simplifies to |x - 5| < epsilon/3. So delta = epsilon/3 works
The epsilon-delta definition also proves the Limit Laws. For instance, the Sum Law follows by splitting the error tolerance in half: require each function to be within epsilon/2 of its limit, then the sum is within epsilon of the sum of limits
The Squeeze Theorem and Substitution Theorem for limits can also be proved rigorously from the epsilon-delta definition
f(x) is continuous at x = a when lim(x->a) f(x) = f(a). Three conditions: f(a) is defined, the limit exists, and they match
Types of discontinuity:
Removable: the limit exists but f(a) is missing or wrong (a hole in the graph)
Jump: left and right limits exist but disagree (a step in the graph)
Vertical asymptote: f(x) blows up to infinity
Oscillation: the limit does not exist at all
Almost all functions built from standard formulas are continuous wherever they are defined. Discontinuities happen where the formula breaks down (division by zero, square root of a negative number)
Composition preserves continuity: if g is continuous at a, and f is continuous at g(a), then f(g(x)) is continuous at a
If f is continuous on [a, b] and r is between f(a) and f(b), then f(c) = r for some c in (a, b)
Use case: proving a solution exists without finding it explicitly
Example: show cos(x) = x has a solution. Let f(x) = cos(x) - x. Then f(0) = 1 > 0 and f(pi) = -1 - pi < 0. Since f is continuous, the IVT guarantees some c in (0, pi) with f(c) = 0
The IVT does not apply to discontinuous functions. A function with a vertical asymptote can jump past a value without ever taking it
v = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}m_{\text{tangent}} = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}Tangent line equation: y = f(a) + m(x - a)
Limit Laws (all require that the individual limits exist):
lim [f +/- g] = lim f +/- lim g
lim [f * g] = lim f * lim g
lim [f / g] = lim f / lim g (denominator limit nonzero)
lim [f]^n = [lim f]^n
Squeeze Theorem: f(x) <= g(x) <= h(x) near a, with lim f = lim h = L, implies lim g = L.
Epsilon-delta: lim(x->a) f(x) = L iff for all epsilon > 0 there exists delta > 0 such that 0 < |x - a| < delta implies |f(x) - L| < epsilon.
Continuity condition: lim(x->a) f(x) = f(a).
IVT: f continuous on [a, b], r between f(a) and f(b), implies there exists c in (a, b) with f(c) = r.
Instantaneous velocity is the most direct application: GPS units, speedometers, and physics simulations all compute rates of change that are, underneath, limit calculations. Any time you see "how fast is something changing right now" rather than "how fast did it change on average," you are looking at a limit.
The Intermediate Value Theorem underpins root-finding algorithms in engineering and computer science. If a bridge's stress function is positive at one load and negative at another, the IVT guarantees a critical failure point exists in between, even before you calculate where exactly it is.
Students often think lim(x->a) f(x) must equal f(a). It does not. The limit depends only on values of f(x) for x near a, not at a itself. A function can have a limit of 5 at x = 3 even if f(3) = 7 or f(3) is undefined.
Students often think "the limit does not exist" and "the limit is infinity" mean the same thing. They do not. Saying the limit is infinity is a specific description of how the function behaves (it grows without bound). "Does not exist" is broader and includes cases like oscillation where the function has no single trend.
Students often think you can always find a limit by plugging in nearby values on a calculator. This can be misleading. The function sin(pi/x) gives 0 at x = 1/2, 1/3, 1/4, ... but the limit as x->0 does not exist. Numerical evidence can deceive.
Students often confuse "continuous" with "smooth." The function f(x) = |x| is continuous everywhere but has a sharp corner at x = 0. Continuity does not require the graph to be smooth.
⚠️ Evaluating limits by factoring and cancelling 0/0 indeterminate forms is tested heavily. Practise polynomial factoring and conjugate-radical tricks until they are automatic.
⚠️ One-sided limits and determining whether a two-sided limit exists: exam questions frequently give a piecewise function and ask for left limit, right limit, and whether the limit exists.
⚠️ Vertical asymptotes: you will be asked to find them and to determine the sign (positive or negative infinity) on each side. This requires sign analysis of the factored form.
⚠️ Epsilon-delta proofs for linear functions (like 3x - 2) appear on exams as "show that" or "prove that" problems. Know the template: solve |f(x) - L| < epsilon for |x - a| < delta.
⚠️ IVT application problems: "show that a solution exists on the interval [a, b]" requires checking continuity, computing f(a) and f(b), confirming the target value lies between them, then citing the theorem.
True or false: lim(x->2) f(x) can exist even if f(2) is undefined. (True.)
Fill in the blank: If lim(x->a+) f(x) = 3 and lim(x->a-) f(x) = 5, then lim(x->a) f(x) = ____. (Does not exist, because the one-sided limits disagree.)
True or false: lim(x->0) 1/x = infinity. (False. The function goes to +infinity from the right and -infinity from the left, so the two-sided limit does not exist as infinity.)
Fill in the blank: To resolve the indeterminate form 0/0 in lim(x->3) (x^2 - 9)/(x - 3), you should ____ the numerator and cancel. (Factor.)
True or false: The IVT can be used to prove a root exists even for a function with a vertical asymptote on the interval. (False. The IVT requires continuity on the closed interval.)
Q: Evaluate lim(x->4) (x^2 - 16) / (x - 4).
A: Factor the numerator: (x-4)(x+4)/(x-4) = x+4. Substituting x = 4 gives 8.
Q: Evaluate lim(x->0) x * sin(1/x).
A: Since -1 <= sin(1/x) <= 1, we have -|x| <= x sin(1/x) <= |x|. Both bounds go to 0, so by the Squeeze Theorem the limit is 0.
Q: For the function f(x) = (2x - 4) / (x^2 - 4x + 3), find all vertical asymptotes and determine the sign of f(x) on each side.
A: Factor: f(x) = 2(x-2) / [(x-1)(x-3)]. Vertical asymptotes at x = 1 and x = 3 (denominator zero, numerator nonzero). Sign analysis: for x just below 1, all three factors (x-2), (x-1), (x-3) are negative, so f(x) = 2(-)/((-)(-)]) = negative, hence lim(x->1-) f(x) = -infinity. For x just above 1, (x-1) flips to positive, giving f(x) = positive, hence lim(x->1+) f(x) = +infinity. Similar analysis at x = 3.
Q: Use the epsilon-delta definition to prove lim(x->5) (3x - 2) = 13.
A: Given epsilon > 0, choose delta = epsilon/3. If 0 < |x - 5| < delta, then |(3x - 2) - 13| = |3x - 15| = 3|x - 5| < 3(epsilon/3) = epsilon.
Q: Show that the equation x^3 + x - 1 = 0 has a solution in (0, 1).
A: Let f(x) = x^3 + x - 1. Then f(0) = -1 < 0 and f(1) = 1 > 0. Since f is a polynomial and therefore continuous, the IVT guarantees some c in (0, 1) with f(c) = 0.
Limits feed directly into the definition of the derivative (Chapter 2). Every derivative is a limit of a difference quotient, so the factoring and cancelling techniques you learn here come back immediately. The epsilon-delta framework also reappears when proving derivative rules and in Real Analysis courses.
Continuity is a prerequisite for the Mean Value Theorem and the Fundamental Theorem of Calculus, both of which require the function to be continuous on a closed interval. The IVT is also the theoretical basis for bisection and Newton's Method (Chapter 3).
limit definition, evaluating limits, limit laws, one-sided limits, infinite limits, vertical asymptotes, Squeeze Theorem, Sandwich Theorem, Pinching Theorem, epsilon-delta, continuity, continuous function, removable discontinuity, jump discontinuity, IVT, Intermediate Value Theorem, root-finding, indeterminate form 0/0, factoring limits, rationalising, conjugate radical, secant line, tangent slope, instantaneous velocity, average velocity, difference quotient, MATH 132, Calculus I, Stewart Chapter 1