First-year courses · Mathematics

Calculus I: limits and continuity

Flashcards on limits, one-sided and infinite limits, the squeeze theorem, continuity, the IVT and asymptotes, with worked examples for first-year calculus.

39 cards6 topicsAbout 15 minutes a roundTypical first-year Calculus I (differential calculus), opening unit on limits and continuity

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About this deck

This deck covers the first few weeks of a first-year differential calculus course: what a limit means, how to evaluate limits with algebra (factoring, rationalizing, the squeeze theorem), one-sided and infinite limits, limits at infinity, the three kinds of discontinuity, the Intermediate Value Theorem, and vertical, horizontal and slant asymptotes. Almost every later idea in calculus, including the derivative and the definite integral, is defined as a limit, so time spent here pays off all term.

Before you start you should be comfortable factoring polynomials, simplifying rational expressions, working with radicals and conjugates, and reading the graphs of basic functions (polynomials, 1/x, sqrt(x), e^x, ln x, sin x and cos x). Radian measure is assumed throughout.

To study, read the term, then work the example on paper before you flip the card; saying the answer aloud is not enough for limit questions because the algebra is where marks are lost. Shuffle across tags once you know each group, since tests rarely tell you which technique to use. A good habit is to sketch a quick graph for every one-sided limit card.

Common mistakes: treating 0/0 as if it were 0 or 1 instead of a signal to simplify; claiming a limit exists when the left and right limits differ; and confusing f(a) with the limit as x approaches a (a function can have a limit at a point where it is undefined). Also remember that writing a limit equals infinity describes how the limit fails to exist; it is not a numerical value.

Before you start: Grade 12 advanced functions (or Math 30-1 / Pre-Calculus 12) algebra, including factoring, rational expressions, logarithms and trigonometry in radians.

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Cover the definitions with a sheet of paper and test yourself, or use the study modes above.

Limit basics

Limit of a function at a point: The single number L that f(x) gets arbitrarily close to as x gets close to a from both sides, whatever value f(a) has, or even if f(a) is undefined.Example: As x approaches 2, (x^2 - 4)/(x - 2) approaches 4, even though the expression is undefined at x = 2.
Left-hand limit: The value f(x) approaches as x moves toward a using only inputs smaller than a. Written lim as x approaches a- of f(x).Example: For f(x) = |x|/x, the value from the left at 0 is -1.
Right-hand limit: The value f(x) approaches as x moves toward a using only inputs larger than a. Written lim as x approaches a+ of f(x).Example: For f(x) = |x|/x, the value from the right at 0 is 1.
When a two-sided limit exists: It exists exactly when both one-sided limits exist (as finite numbers) and are equal; the common value is the limit.Example: |x|/x has left value -1 and right value 1 at x = 0, so the two-sided limit does not exist.
Epsilon-delta definition: lim f(x) = L as x approaches a means: for every epsilon > 0 there is a delta > 0 such that 0 < |x - a| < delta guarantees |f(x) - L| < epsilon.Example: For f(x) = 3x, a = 1, L = 3: |3x - 3| = 3|x - 1| < epsilon whenever |x - 1| < epsilon/3, so choose delta = epsilon/3.
Direct substitution property: If f is a polynomial, or a rational function whose denominator is not zero at a, the limit as x approaches a is simply f(a).Example: As x approaches 3, x^2 + 1 approaches 3^2 + 1 = 10.
Limit that fails by oscillation: A limit can fail to exist because the function keeps swinging between values instead of settling down, even though it stays bounded.Example: sin(1/x) takes every value between -1 and 1 in every interval around 0, so it has no limit as x approaches 0.
Limits of piecewise functions: At a break point, compute the left limit with the piece used for x < a and the right limit with the piece used for x > a, then compare.Example: f(x) = x + 1 for x < 1 and 3x for x >= 1: left value 2, right value 3, so no two-sided limit at x = 1.

Limit laws and techniques

Sum, difference and constant multiple laws: If lim f = L and lim g = M (both finite), then lim (f + g) = L + M, lim (f - g) = L - M and lim cf = cL.Example: If lim f = 4 and lim g = -1 as x approaches 2, then lim (3f - g) = 12 + 1 = 13.
Product and quotient laws: With lim f = L and lim g = M finite, lim fg = LM, and lim f/g = L/M provided M is not 0.Example: If lim f = 6 and lim g = 3, then lim f/g = 2.
Indeterminate form 0/0: Direct substitution gives zero over zero; this says nothing about the answer and means more algebra (or another method) is needed.Example: (x^2 - 9)/(x - 3) at x = 3 gives 0/0, but the limit is 6 after simplifying.
Factor and cancel technique: Factor numerator and denominator, cancel the common factor that causes 0/0 (allowed because x is not equal to a), then substitute.Example: (x^2 - 1)/(x - 1) = x + 1 for x not equal to 1, so the limit as x approaches 1 is 2.
Rationalizing with a conjugate: Multiply top and bottom by the conjugate of an expression containing a square root to remove the 0/0, then simplify and substitute.Example: (sqrt(x + 9) - 3)/x = 1/(sqrt(x + 9) + 3) after multiplying by the conjugate, so the limit at 0 is 1/6.
Squeeze theorem: If g(x) <= f(x) <= h(x) near a, and g and h both approach the same number L at a, then f must also approach L at a.Example: -x^2 <= x^2 sin(1/x) <= x^2, and both bounds go to 0, so x^2 sin(1/x) approaches 0 as x approaches 0.
Limit of sin(x)/x at 0: As x approaches 0 (x in radians), the ratio of sin x to x approaches 1. It is proved with the squeeze theorem and a unit-circle area argument.Example: sin(5x)/x = 5 sin(5x)/(5x), which approaches 5 * 1 = 5.
Limit of (1 - cos x)/x at 0: This ratio approaches 0 as x approaches 0 (radians). Together with sin(x)/x approaching 1, it is used to differentiate sine and cosine from first principles.Example: Multiply by (1 + cos x): the expression becomes sin^2(x)/(x(1 + cos x)), which approaches 1 * 0/2 = 0.
Limit of a composition: If g(x) approaches b and f is continuous at b, then f(g(x)) approaches f(b); the limit can move inside a continuous outer function.Example: As x approaches 0, x^2 + pi approaches pi, so cos(x^2 + pi) approaches cos(pi) = -1.

Infinite limits and limits at infinity

Infinite limit: f(x) grows without bound (positive or negative) as x approaches a. Writing that the limit equals infinity describes the behaviour; the limit is not a real number.Example: As x approaches 0, 1/x^2 grows without bound, so we write the limit is infinity.
Limit at infinity: The value f(x) settles toward as x grows without bound in the positive direction (or the negative direction).Example: 1/x approaches 0 as x approaches infinity.
Rational functions at infinity: Compare degrees: equal degrees give the ratio of leading coefficients; a smaller top degree gives 0; a larger top degree gives no finite limit.Example: (3x^2 + 1)/(5x^2 - x) approaches 3/5 as x approaches infinity.
Dividing by the highest power: To find a limit at infinity, divide every term by the highest power of x in the denominator so most terms visibly go to 0.Example: (2x + 7)/(x^2 + 1): dividing by x^2 gives (2/x + 7/x^2)/(1 + 1/x^2), which approaches 0.
Exponential and log limits: e^x approaches 0 as x approaches negative infinity and grows without bound as x approaches infinity; ln x goes to negative infinity as x approaches 0 from the right.Example: e^(-x) approaches 0 as x approaches infinity, so y = 0 is a horizontal asymptote of e^(-x).

Continuity

Continuity at a point: f is continuous at a when three things hold: f(a) is defined, the limit as x approaches a exists, and that limit equals f(a).Example: f(x) = x^2 is continuous at 3 because the limit and the value are both 9.
Removable discontinuity: A break where the limit exists but f(a) is undefined or differs from it; redefining one point would make the function continuous (a hole in the graph).Example: (x^2 - 1)/(x - 1) has a hole at x = 1; defining f(1) = 2 removes it.
Jump discontinuity: A break where both one-sided limits exist as finite numbers but are not equal, so the graph jumps.Example: The floor function jumps from 0 to 1 at x = 1.
Infinite discontinuity: A break where at least one one-sided limit is infinite; the graph has a vertical asymptote there.Example: 1/(x - 2) has an infinite discontinuity at x = 2.
Continuity on an interval: Continuous at every interior point of the interval; at a closed endpoint only the one-sided limit from inside the interval is required to match.Example: sqrt(x) is continuous on [0, infinity) using the right-hand limit at 0.
Functions continuous on their domains: Polynomials, rational functions, roots, exponentials, logarithms and the trig functions are continuous at every point where they are defined.Example: tan x is continuous everywhere except x = pi/2 + k pi, where it is undefined.
Combining continuous functions: Sums, differences, products and compositions of continuous functions are continuous, and so are quotients wherever the denominator is not 0.Example: e^(sin x) is continuous everywhere because e^x and sin x are.
Choosing a constant for continuity: To make a piecewise function continuous at a break point, set the left limit, right limit and function value equal and solve for the unknown constant.Example: f(x) = x^2 for x < 2 and kx for x >= 2: need 4 = 2k, so k = 2.

Key theorems

Intermediate Value Theorem (IVT): If f is continuous on [a, b] and N is any number between f(a) and f(b), then f(c) = N for at least one c in (a, b).Example: A continuous temperature that goes from 10 to 20 degrees must pass through 15 degrees at some moment.
Using the IVT to show a root exists: Find a closed interval where a continuous function changes sign; zero lies between the endpoint values, so there is a root inside.Example: f(x) = x^3 + x - 1: f(0) = -1 and f(1) = 1, so there is a root between 0 and 1.
Bisection method: Repeatedly halve an interval on which a continuous function changes sign, keeping the half where the sign change persists, to trap a root.Example: For x^3 + x - 1, f(0.5) = -0.375 < 0 and f(1) > 0, so the root is in (0.5, 1).
Why continuity matters for the IVT: Without continuity on the whole closed interval, a sign change does not guarantee a root because the graph can jump over zero.Example: 1/x is -1 at x = -1 and 1 at x = 1 but is never 0; it is not continuous at 0.

Asymptotes

Vertical asymptote: A line x = a where f(x) grows without bound (up or down) as x approaches a from at least one side.Example: (x + 1)/(x - 3) has a vertical asymptote at x = 3.
Horizontal asymptote: A line y = L where f(x) approaches L as x approaches infinity or negative infinity. A graph may cross it for finite x.Example: (2x + 1)/(x - 4) approaches 2, so y = 2 is a horizontal asymptote.
Slant (oblique) asymptote: A non-horizontal line y = mx + b that the graph approaches at infinity; for rational functions it occurs when the top degree is one more than the bottom.Example: (x^2 + 1)/x = x + 1/x, so y = x is a slant asymptote.
Hole versus vertical asymptote: If a factor cancels from top and bottom, its zero gives a hole; zeros of the denominator that remain after cancelling give vertical asymptotes.Example: (x - 2)/(x^2 - 4) = 1/(x + 2) for x not 2: hole at x = 2, vertical asymptote at x = -2.
Asymptote of ln x: The natural logarithm drops without bound as x approaches 0 from the right, so the line x = 0 is a vertical asymptote of its graph.Example: ln(0.001) is about -6.9 and ln(0.000001) is about -13.8.