Unit notes

Function notation f(a) means the output at input a, not multiplication. Factoring changes the form of an expression without changing its value where both forms are defined. An open circle on a graph excludes that point; it does not prevent nearby function values from approaching its height.

The limit of f(x) as x approaches a describes output behavior for inputs close to a, not necessarily at a. The function value f(a) can differ from the limit or be undefined. A two-sided limit exists only when both one-sided limits exist and agree.

Tables and graphs provide evidence, but a finite display can hide behavior close to the target. Evaluate from both sides and consider the expression’s structure. A limit of positive infinity describes unbounded growth; it is not a finite real-valued limit. Oscillation can also prevent a limit from existing.

For continuous expressions at an allowed input, direct substitution gives the limit. A result of 00 is indeterminate: it does not mean the limit is zero, one, or nonexistent. Factor and cancel a common factor on nearby allowed inputs, rationalize, or combine fractions as appropriate.

Cancellation changes the expression only on the domain where the canceled factor is nonzero. That is enough for a nearby limit but does not define the original function at the excluded input. Keep domain restrictions while simplifying. Standard trigonometric limits use radian measure.

A vertical asymptote involves unbounded behavior near a finite input. A horizontal asymptote describes a finite limit as x goes to positive or negative infinity. A graph may cross a horizontal asymptote because the statement concerns far-away behavior.

For rational functions, compare leading powers for limits at infinity. For a vertical asymptote, inspect signs on each side; opposite signs can give opposite infinities. Dividing by a small quantity does not determine the sign without considering numerator and denominator.

A function is continuous at a when f(a) is defined, the two-sided limit exists, and that limit equals f(a). At a domain endpoint, use the relevant one-sided continuity. A removable discontinuity can be repaired by assigning the limiting value, while a jump cannot be fixed by changing just one point.

For a piecewise function, match one-sided limits at the joining input and check the assigned value. A continuous graph can still have a corner or vertical tangent, so continuity alone does not guarantee differentiability.

If f is continuous on [a,b], every value between f(a) and f(b) is attained somewhere in the interval. Opposite-sign endpoint values therefore guarantee at least one zero. State continuity and the endpoint comparison before invoking the theorem.

The theorem does not identify an exact location or prove uniqueness. A discontinuous function can jump over a value, making the hypothesis essential. A squeeze argument is different: bound a function between two others with the same limit to establish its limiting behavior.

First determine what the input approaches: a finite number, positive infinity, or negative infinity. For a finite input, test direct substitution when the formula is continuous there. A numerical result gives the limit. The form 00 is not an answer; it means the first substitution test has not determined the limit. Look for a common factor, a conjugate, or a trigonometric identity that changes the expression on a punctured neighborhood of the input.

For example, (x²−9)(x−3) equals x+3 only when x≠3. That is enough to determine the limit as x approaches 3, because a limit concerns nearby inputs. It does not define the original function at 3. When using a table, choose values from both sides and move progressively closer. A table suggests behavior but can miss oscillation or a discontinuity between sample points. Algebra and theorems provide the justification.

A two-sided limit exists as a finite number only if the left-hand and right-hand limits both exist and agree. The actual value assigned at the boundary can be different. For a piecewise formula, use the expression valid to the left for the left limit and the expression valid to the right for the right limit. Do not select a branch merely because it contains an equality sign.

To make a piecewise function continuous, set the two limiting expressions equal and ensure the defined function value equals the common limit. To identify a vertical asymptote, examine whether function values grow without bound near the input. A denominator equal to zero is only a candidate: a factor might cancel and leave a removable hole. Use signs on either side to distinguish positive and negative infinite behavior.

The Intermediate Value Theorem requires continuity on a closed interval. If a target value lies between the endpoint outputs, at least one input in that interval reaches the target. A sign change is the special case used to guarantee a zero. Write the continuity condition and the endpoint comparison before stating the conclusion. The theorem guarantees existence, not uniqueness or an exact location.

A graph can have a jump over a target value, so endpoint values alone are insufficient. Likewise, a continuous function may hit the same target many times. When a question asks for a justification, “because the graph crosses” is weaker than explaining why continuity and the endpoint values force a crossing. Do not use a calculator approximation as a substitute for the theorem’s hypotheses.

For position s(t), average velocity from t=a to t=a+h is [s(a+h)−s(a)]h, provided h≠0. The numerator is a change in position and the denominator is a change in time. Units matter: meters divided by seconds gives meters per second. On a position graph, this quotient is a secant slope.

To investigate an instantaneous rate, examine what the average rate approaches as h approaches zero. We do not begin by setting h=0, because the original quotient would be undefined. A limit describes a nearby pattern that may remain meaningful even where a particular formula cannot be evaluated.

For s(t)=t² at t=3, the average rate is [(3+h)²−9]h. Expanding and canceling for h≠0 gives 6+h. Thus values from both positive and negative h approach 6. This motivates a derivative, but this unit focuses on the limit reasoning behind that conclusion.

The statement lim as x→a of f(x)=L means that f(x) approaches L as x gets arbitrarily close to a, considering x values in the domain near a and excluding a itself. The input approaches a; the output approaches L. These numbers need not be equal.

Changing only f(a) leaves the nearby values unchanged, so it cannot change an existing limit. A graph may have an open circle at the approached height and a filled point at another height. The filled point tells you f(a), while the surrounding curve tells you the limit.

For an ordinary two-sided finite limit at an interior point, the left and right approaches must agree. A function value can exist even when this agreement fails. Conversely, a limit can exist when the function is undefined at the point.

The notation x→a^- means inputs less than a approach a; x→a⁺ means inputs greater than a approach a. It does not say whether outputs are positive or negative. A left-hand limit can be positive, and a right-hand limit can be negative.

If the left-hand limit is L and the right-hand limit is the same L, then the two-sided limit is L. If they approach distinct values, the two-sided limit does not exist. Taking their average invents a number that does not describe either approach.

For piecewise functions, nearby inputs on each side may follow different formulas. Substitute the boundary into each applicable branch only after identifying the side. The formula assigned at the boundary determines the point value but does not repair a jump between the branches.

When f(x)→L and g(x)→M are finite, the limit of f+g is L+M and the limit of fg is LM. Constant multiples and powers behave similarly. The quotient rule for limits requires M≠0. A denominator tending to zero means the quotient rule cannot immediately give a finite answer.

Polynomials are continuous at every real input, so their limits can be found by evaluation. Rational functions are continuous where the denominator is nonzero. Roots and compositions require attention to domain and continuity at the relevant values; symbols alone are not permission to substitute.

If substitution yields an ordinary defined number, it may finish the problem. If it yields 00, you have an indeterminate form: nearby behavior could lead to different limits. If a nonzero numerator is divided by something tending to zero, investigate signs and one-sided unbounded behavior.

If substitution into a rational expression gives 00, try factoring. A common nonzero factor can be canceled for nearby inputs, but the cancellation does not restore an excluded input to the original domain. Limits allow this because they examine inputs near, rather than exactly at, the target.

For (x²−9)(x−3), the numerator factors into (x−3)(x+3). For x≠3 the expression equals x+3. In contrast, you cannot cancel x from (x+3)x by crossing out the x terms: the numerator is a sum, not a product containing a common factor.

Not every 00 form is resolved by factoring, and not every expression with a canceled factor has the same domain. State the restriction, simplify the nearby expression and then evaluate its limit. This sequence makes the reasoning visible.

For a difference of square roots, multiply numerator and denominator by the conjugate. The identity (A−B)(A+B)=A²−B² removes the radicals from that product. This often creates a common factor that can be canceled for nearby, permitted inputs.

The standard limit sin xx→1 as x→0 uses radian measure. To evaluate sin(3x)x, write it as 3·sin(3x)(3x). The inner input 3x approaches zero, so the standard limit applies and the result is 3. Do not forget the factor introduced by matching the denominator.

Choose a transformation based on the obstacle: factoring for polynomial cancellation, a conjugate for radical differences, or a known limit after a substitution. A collection of algebraic steps is useful only if it produces an expression whose nearby behavior is easier to justify.

Continuity at x=a requires that f(a) be defined, that the two-sided limit as x→a exist, and that the limit equal f(a). Checking only one or two conditions is insufficient. The equality combines a statement about nearby behavior with a statement about the exact point.

A removable discontinuity has a finite limit but an undefined or mismatched point value. A jump has differing finite one-sided limits. Infinite or oscillatory behavior can prevent a finite limit as well. A one-point reassignment can repair a removable discontinuity, but it cannot force disagreeing nearby branches to match.

Continuity on a closed interval uses right continuity at the left endpoint and left continuity at the right endpoint. Inside the interval, use ordinary two-sided continuity. Domain restrictions remain important: a rational function is continuous on portions of its domain, not automatically across excluded denominator zeros.

If each formula is continuous on its own interval, the main issue is the boundary. Evaluate the left-hand limit using the left branch, the right-hand limit using the right branch, and the function value using whichever branch includes equality. Continuity requires all three values to coincide.

For f(x)=kx+1 when x<2 and f(x)=x² when x≥2, the left limit is 2k+1, while the right limit and f(2) equal 4. Solving 2k+1=4 gives k=32. The equation is not arbitrary: it expresses equality of the two approaches.

Joining the heights does not necessarily join the slopes. A corner can be continuous. Derivative conditions come later; do not add unnecessary slope requirements when a problem asks only for continuity.

If f is continuous on [a,b], it attains every output between f(a) and f(b) at some input in that interval. For a target strictly between the endpoint outputs, at least one such input lies in (a,b). To justify a zero, continuity plus opposite signs at the endpoints is a common application.

The theorem guarantees at least one matching input, not exactly one, and does not tell you its value. A function may rise and fall and cross the same output repeatedly. Extra information, such as strict monotonicity, can support uniqueness, but is not supplied merely by continuity.

A function taking −1 for x<0 and 1 for x≥0 changes sign without ever equaling zero. This shows why endpoint signs alone are insufficient. State continuity on the entire closed interval and bracket the target before invoking the theorem.

Writing f(x)→+∞ means outputs eventually exceed any chosen positive bound as inputs approach the target in the specified way. It is not an ordinary finite function value. A statement that the finite limit does not exist can be made more informative by describing the one-sided unbounded behavior.

For 1(x−2), the denominator is negative just left of 2 and positive just right of 2. Its magnitude becomes small, so the quotient goes toward −∞ on the left and +∞ on the right. For 1(x−2)², the squared denominator is positive on both sides, so both approaches go toward +∞.

A vertical asymptote occurs when at least one one-sided limit is unbounded. A zero denominator alone does not prove one: a common factor might cancel and leave a removable discontinuity. Factor and inspect nearby behavior before classifying the break.

In a limit as x→∞, inputs become arbitrarily large; they are not approaching a finite vertical asymptote. To analyze a rational function, compare the highest powers. Dividing numerator and denominator by the highest denominator power often shows which terms vanish.

For equal-degree polynomials, the limit at positive or negative infinity is the ratio of leading coefficients. If the numerator degree is smaller, the limit is zero. If it is larger, the function may grow without bound; the sign and direction require analysis, not just a degree count.

The identity √(x²)=|x| matters when x→−∞. For example, x√(x²+1) approaches 1 as x→∞ and −1 as x→−∞. A horizontal asymptote describes end behavior. The graph can cross it at finite inputs without invalidating that end behavior.

Identify the target and domain. Try substitution when continuity permits. If it gives a defined value, finish with that justification. If it gives 00, look for factoring, a conjugate, a standard limit or a bounding argument. If a nonzero quantity is divided by something approaching zero, inspect one-sided signs.

For a piecewise boundary, use the branch for each approach. For inputs tending to infinity, study dominant terms rather than treating infinity as a number to substitute. For a squeeze problem, check both the inequality and the bounding limits.

Estimate a nearby value to detect an implausible sign or magnitude, but do not replace your justification with the estimate. Keep excluded inputs in view. Use radian measure for the standard trigonometric limit. State whether you found a finite limit, an infinite behavior, or nonexistence through disagreement.