The two-sided limit exists exactly when both one-sided limits exist and agree on a single value. This converts the problem of evaluating a two-sided limit into two simpler problems — compute each direction separately, then check whether they match.
The identity function returns its input unchanged. As x approaches a, the output also approaches a. Combined with the constant rule, this provides the base case for evaluating limits of all polynomial expressions.
The limit of a sum is the sum of the limits; the limit of a difference is the difference of the limits. Limits distribute over addition and subtraction whenever the component limits exist.
Constants pass through limits. Scaling a function by a constant scales its limit by the same constant. This is a special case of the product rule with one factor constant, but it appears often enough to state on its own.
The limit of a product is the product of the limits. The rule extends to any finite number of factors: if every factor has a limit, the product's limit is the product of those limits.
The limit of a quotient is the quotient of the limits, provided the denominator's limit is nonzero. When the denominator's limit is zero, this rule fails and other techniques are required.
The limit of a power is the power of the limit. For positive integer n this follows from repeated application of the product rule. The rule extends to rational exponents under domain restrictions.
Absolute value passes through limits. The converse is false: lim∣f∣ may exist when limf does not — for instance, ∣(−1)n∣=1 for all n, but (−1)n has no limit.
For any polynomial, the limit at a point equals the value at that point. Direct substitution always works. This follows from polynomials being continuous everywhere — every polynomial is built from sums, products, and constant multiples of the identity function and constants, all operations that limits respect.
When the denominator does not vanish at a, the limit of a rational function is just the value at a — direct substitution works. When q(a)=0, this rule no longer applies: the result is either an infinite limit (if p(a)=0) or an indeterminate 00 (if p(a)=0, indicating a shared factor of (x−a)).
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Composition Rule (Limits)
x→alimf(g(x))=f(x→alimg(x))if f is continuous at x→alimg(x)
x→alimf(g(x))=f(x→alimg(x))if f is continuous at x→alimg(x)
Limits pass through continuous functions. First find the limit of the inner function, then apply the outer function to that value. The result equals the limit of the composition.
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Squeeze Theorem
If g(x)≤f(x)≤h(x) near a and x→alimg(x)=x→alimh(x)=L,
then x→alimf(x)=L.
If g(x)≤f(x)≤h(x) near a and x→alimg(x)=x→alimh(x)=L,
When a function is trapped between two others that converge to the same limit, it has nowhere to go but that limit. The Squeeze Theorem proves the foundational trigonometric limit limx→0xsinx=1 by bounding the ratio between cosx and 1 near zero.
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L'Hôpital's Rule
x→alimg(x)f(x)=x→alimg′(x)f′(x)for indeterminate forms 00 or ∞∞
x→alimg(x)f(x)=x→alimg′(x)f′(x)for indeterminate forms 00 or ∞∞
When direct substitution gives 00 or ∞∞, replace numerator and denominator with their derivatives and try again. The new limit, if it exists, equals the original. The rule may need to be applied repeatedly when the indeterminate form persists.
A single equation that encodes three requirements: f(a) must be defined, the limit must exist, and the two must match. Continuity means the function value matches what surrounding values predict — no jumps, no holes, no surprises.
Continuity from a single direction. A function continuous on a closed interval [a,b] must be continuous on (a,b), right-continuous at a, and left-continuous at b — full continuity is unavailable at endpoints because only one direction of approach exists.
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Continuity Preserved Under Operations
f,g continuous at a⟹f±g,cf,f⋅g,gf(g(a)=0),f∘g continuous at a
f,g continuous at a⟹f±g,cf,f⋅g,gf(g(a)=0),f∘g continuous at a
Continuity is preserved by the standard operations — sums, differences, scalar multiples, products, quotients (where defined), and compositions. This means whole families of functions are continuous wherever defined: polynomials everywhere, rational functions where the denominator is nonzero, and any composition built from continuous building blocks.
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Intermediate Value Theorem
f continuous on [a,b],k between f(a) and f(b)⟹∃c∈(a,b) with f(c)=k
f continuous on [a,b],k between f(a) and f(b)⟹∃c∈(a,b) with f(c)=k
A continuous function on a closed interval hits every value between its endpoints — no skipping. If f is negative at one endpoint and positive at the other, it must equal zero somewhere in between. This is an existence theorem: it guarantees a c exists, but says nothing about where it is or whether it is unique.
Direct substitution gives 00, but the geometry of the unit circle reveals that for small x, sinx and x are nearly equal. Their ratio approaches exactly 1. This limit is the foundation for the derivatives of sinx and cosx.
Another 00 form. Multiply by the conjugate 1+cosx1+cosx to convert the numerator to sin2x, then split into xsinx⋅1+cosxsinx — the first factor goes to 1, the second to 0.
The numerator vanishes like x2 near zero — specifically, 1−cosx≈2x2. The leading coefficient of that approximation is exactly the limit. The second derivative of cosx at zero is −1, so the Taylor expansion gives cosx≈1−2x2.
This limit is the derivative of ex at x=0, and it is exactly what makes e special — it is the unique base for which the slope at zero equals 1. The result drives the entire structure of ex as the function equal to its own derivative.
The mirror image of the exponential limit at zero: ln(1+x) behaves like x near zero. Substituting u=ln(1+x) converts this directly into the exponential limit. Equivalently, this is the derivative of lnx at x=1, which equals 1.
The number e≈2.71828 is defined by this limit. It arises in continuous compounding: an annual interest rate of 100% compounded n times per year produces a growth factor (1+1/n)n, which approaches e as compounding becomes continuous.
An indeterminate 0⋅(−∞) form. As x→0+, lnx plunges to −∞, but x vanishes faster than lnx blows up. The polynomial factor wins; the product approaches zero.
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Asymptotes & End Behavior
(6 formulas)
Horizontal Asymptote Condition
x→∞limf(x)=L or x→−∞limf(x)=L⟹y=L is a horizontal asymptote
x→∞limf(x)=L or x→−∞limf(x)=L⟹y=L is a horizontal asymptote
A finite limit at infinity in either direction produces a horizontal asymptote. A function can have zero, one (the same in both directions), or two (different finite limits at ±∞) horizontal asymptotes.
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Vertical Asymptote Condition
x→a−limf(x)=±∞ or x→a+limf(x)=±∞⟹x=a is a vertical asymptote
x→a−limf(x)=±∞ or x→a+limf(x)=±∞⟹x=a is a vertical asymptote
Any one-sided infinite limit produces a vertical asymptote. Both sides can disagree in sign — the line x=a is still a vertical asymptote even when one side goes to +∞ and the other to −∞. For rational functions, vertical asymptotes typically occur where the denominator vanishes while the numerator does not.
The exponential grows without bound to the right and decays to zero to the left. The horizontal asymptote y=0 appears in the direction where the exponent goes to −∞. For e−x, the directions reverse.
The natural logarithm grows without bound (slowly) and plunges to −∞ as the argument approaches zero from the right. The line x=0 is a vertical asymptote. No left-hand limit at zero exists because lnx is undefined for x≤0.
Any exponential eventually overtakes any polynomial. Even when the polynomial degree is enormous, the exponential's rate of growth wins for large enough x. This is the upper tier of the growth-rate hierarchy.
Any positive power of x eventually overtakes any logarithm. Logarithms grow but slowly — slower than every polynomial of positive degree. This is the bottom tier of the growth-rate hierarchy.
Differentiating an accumulation function recovers the integrand. If you accumulate f from a up to a moving boundary x, the rate at which the accumulated total grows at x is exactly f(x). This guarantees that every continuous function has an antiderivative — namely, its own accumulation function — even when no elementary formula expresses it.
The computational engine of integral calculus. Rather than computing limits of Riemann sums, find any antiderivative and evaluate at the endpoints. The constant of integration cancels in the subtraction, so any antiderivative works — different choices of C produce the same definite integral.
Antiderivatives come in families. If F is one antiderivative of f, every antiderivative has the form F(x)+C for some constant C — because the derivative of any constant is zero, vertical shifts of the graph all share the same derivative. Initial conditions like F(0)=3 pin down C.
Integration distributes over addition and subtraction. Complex integrands break into simpler pieces, each integrated separately. Holds for both definite and indefinite integrals.
An integral over [a,c] can be split at any intermediate point b. Essential for piecewise functions where different formulas apply on different subintervals. The point b need not lie between a and c — the rule extends with appropriate sign conventions.
Swapping the bounds of a definite integral negates the result. The Riemann sum construction accumulates contributions in the direction a→b; reversing the direction reverses every signed width Δx, flipping the total's sign.
Pointwise inequality between integrands carries through to integrals. A special case: if f≥0, then ∫f≥0. The comparison property underpins both estimation techniques and convergence tests for improper integrals.
When the integrand is bounded between constants m and M, the integral is bounded between the areas of two rectangles of width b−a and heights m, M. This is the rectangle approximation in its crudest form, and it provides quick sanity checks on computed integrals.
The reverse of the chain rule. When the integrand contains an inner function g(x) multiplied by its derivative g′(x), the substitution u=g(x), du=g′(x)dx collapses the integral into a simpler form in u.
The reverse of the product rule. Splits an integrand into two factors u and dv; differentiating u and integrating dv trades the original integral for a hopefully simpler one. The mnemonic LIATE (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential) ranks function types by how readily they should be chosen as u — earlier types make better choices.
The definite integral computes signed area — regions above the x-axis count positively, regions below count negatively. To get total geometric area regardless of sign, integrate the absolute value of f. In practice this means splitting the interval where f changes sign and integrating each piece with the appropriate sign.
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Standard Antiderivatives — Algebraic & Logarithmic
Increase the exponent by one, then divide by the new exponent. The reverse of the differentiation power rule. The exception n=−1 is critical — that case produces the natural logarithm rather than x0/0, which is undefined.
The exceptional case of the power rule. The absolute value matters: for x>0 the antiderivative is lnx, and for x<0 it is ln(−x) (since differentiating gives 1/x in either case). The two cases collapse into ln∣x∣, valid on either side of zero — but not across it.
When the integrand is a function divided by its own derivative — a "logarithmic derivative" pattern — the antiderivative is the logarithm of the function. This pattern appears constantly in disguise: ∫tanxdx becomes ∫(sinx)/(cosx)dx, which has the form −f′/f with f=cosx.
The natural logarithm has no elementary antiderivative obvious by inspection — it is found via integration by parts with u=lnx and dv=dx. The result xlnx−x is worth memorizing because lnx appears constantly as a multiplicative factor inside more complex integrals.
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Standard Antiderivatives — Exponential, Trig & Inverse Trig
The exponential function ex is its own antiderivative — the same property that makes it its own derivative. This is the defining feature of e: it is the unique base for which the function equals its own rate of change.
For an arbitrary positive base, the antiderivative needs a 1/lna correction factor — this compensates for the chain-rule factor lna that appears when differentiating ax. Setting a=e recovers the natural form, since lne=1.
Rewrite tanx=sinx/cosx, recognize as −f′(x)/f(x) with f=cosx, and apply the logarithmic derivative pattern. Result: −ln∣cosx∣+C, equivalently ln∣secx∣+C.
Less obvious than the other trig integrals. The trick: multiply secx by (secx+tanx)/(secx+tanx) — a clever form of 1. The numerator becomes sec2x+secxtanx, which is exactly the derivative of the denominator secx+tanx. The integrand is now in f′/f form.
Reverses the derivative of arctangent. The general a form is more useful than the special case a=1 — most integrals encountered have an arbitrary constant in place of 1, and completing the square often produces a quadratic of the form a2+(x−h)2 that fits this template.
Reverses the derivative of arcsine. The square root structure a2−x2 also appears as the trigger for trigonometric substitution x=asinθ — the two approaches give the same result.
An odd function f(−x)=−f(x) has graph symmetric through the origin, so the signed area on [−a,0] exactly cancels the signed area on [0,a]. The total integral is zero — without computing anything.
Definite integrals are defined over finite intervals; integration to ∞ is defined as a limit of finite integrals. The improper integral converges if this limit is finite, diverges otherwise. Symmetric definitions handle the −∞ case and integrals with both bounds at infinity (split at any finite point and require both halves to converge).
When the integrand has a vertical asymptote at an endpoint, integrate to a finite cutoff and take the limit as the cutoff approaches the asymptote. For an asymptote at the left endpoint, the limit runs t→a+. For an interior asymptote at c, split the integral at c and take both one-sided limits independently.
The integrals of 1/xp are the canonical benchmarks for comparing other improper integrals. The boundary case p=1 — which gives lnx as an antiderivative — always diverges, both at infinity and at zero. Convergence at infinity requires fast enough decay (p>1); convergence at zero requires slow enough blow-up (p<1).
The continuous analog of an arithmetic mean. The integral computes the total accumulated value, and dividing by the interval length yields the average. Geometrically, favg is the height of the rectangle over [a,b] that has the same area as the region under f. The Mean Value Theorem for Integrals guarantees a continuous f actually attains favg at some point c∈(a,b).
Proved from the limit definition using the angle addition formula and the special limitslimh→0hsinh=1 and limh→0hcosh−1=0. Repeated differentiation cycles with period four: sinx→cosx→−sinx→−cosx→sinx.
The negative sign is essential — a frequent source of error. Follows by differentiating cosx=sin(π/2−x) via the chain rule, or directly from the limit definition using the cosine angle addition formula. Cofunction pattern: cosine, cotangent, and cosecant all carry a negative sign in their derivatives.
The natural exponential is the unique non-trivial function equal to its own derivative — the property that defines e as the natural base. Proved from the limit definition using the special limitlimh→0heh−1=1: the difference quotient factors as ex⋅heh−1, which approaches ex⋅1=ex.
Defined for all real x — no domain restriction. Derived by implicit differentiation of tany=x: sec2y⋅y′=1, and sec2y=1+tan2y=1+x2. Always positive, confirming that arctanx is strictly increasing. As x→±∞, the derivative approaches zero, reflecting the horizontal asymptotes y=±π/2. Reappears prominently in integration as the antiderivative of 1/(1+x2).
Follows directly from the definition sinhx=2ex−e−x together with the derivative of $e^x$: differentiating gives 2ex+e−x=coshx. Hyperbolic derivatives mirror trigonometric derivatives but with no sign flip between sinh and cosh — both differentiate to each other without a negative.
Derived via the quotient rule on tanhx=sinhx/coshx. The hyperbolic identity cosh2x−sinh2x=1 simplifies the numerator to 1, giving 1/cosh2x=sech2x. Mirrors dxd[tanx]=sec2x.
A function with a well-defined tangent line at a must be continuous at a — graphs with jumps or holes cannot have a finite slope at the discontinuity. The converse is false: ∣x∣ is continuous at 0 but has no derivative there (left and right slopes disagree). Continuity is necessary for differentiability but not sufficient.
The implications hold on intervals where the sign is consistent. Follows from the Mean Value Theorem: if f′>0 between a and b, then f(b)−f(a)=f′(c)(b−a)>0.
A critical point is where local extrema can occur — Fermat's theorem states that interior extrema have f′(c)=0 when the derivative exists. But not every critical point is an extremum: f(x)=x3 has a critical point at 0 with neither a max nor a min. Determining the type requires the first or second derivative test. Critical points where f′ is undefined include corners (like ∣x∣ at 0) and vertical tangents (like 3x at 0).
The second derivative measures how the slope is changing. Positive f′′ means the slope is increasing — graph bends upward, lies above its tangent lines. Negative f′′ means the slope is decreasing — graph bends downward, lies below its tangent lines. Sign changes of f′′ identify inflection points.
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Inflection Point Condition
f′′(c)=0 or undefined, and f′′(x) changes sign at c
f′′(c)=0 or undefined, and f′′(x) changes sign at c
An inflection point is where concavity changes. The vanishing (or undefined value) of f′′ is necessary but not sufficient — f(x)=x4 has f′′(0)=0 but no inflection point there, since f′′ stays non-negative on both sides. Confirming an inflection point requires checking that f′′ actually changes sign across c.
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Extreme Value Theorem
f continuous on [a,b]⟹f attains a max and min on [a,b]
f continuous on [a,b]⟹f attains a max and min on [a,b]
Both hypotheses are required: f must be continuous, and the interval must be closed and bounded. A discontinuous f may shoot to infinity; an open interval (a,b) may have f approaching a sup/inf at the missing endpoints but never attaining it.
ex is its own derivative at every order. The unique non-trivial fixed point of differentiation. This invariance makes ex central to differential equations — the equation y′=y has solutions y=Cex.
Each differentiation of eax via the chain rule contributes one factor of a. After n applications, the coefficient is an. Reduces to the natural exponential when a=1.
The derivatives of sinx cycle with period four: sinx→cosx→−sinx→−cosx→sinx. Each differentiation corresponds to a π/2 phase shift — every derivative is just sine evaluated at a shifted argument.
Same period-four cycle as sine, also expressible as a phase shift by π/2 per differentiation. The two patterns together encode the rotational structure of sin and cos under differentiation.
Both x(t) and y(t) must be differentiable, and dx/dt=0 at the point of interest. When dx/dt=0, the tangent line is vertical (assuming dy/dt=0) or the point is singular.