Wemazi

AP Calculus · Unit 1

Limits and Continuity

Limits from graphs, tables and algebra, and continuity.

Topic 1.1Change at an instant

  1. Find change over an intervalUse the difference between the ending and starting amounts, with units.
  2. Calculate an average rateDivide the change in amount by the elapsed input, keeping the sign and units.
  3. Describe change at one instantDistinguish the rate at one moment from an average over an interval.
  4. Use shorter intervals to estimate a rateCompare nearby average rates and identify the instantaneous rate they approach.

Topic 1.2What a limit says

  1. Read what a limit statement meansIdentify the input that x approaches and the output that f(x) approaches.
  2. Compare the two sidesFind the left and right limits and decide whether the two-sided limit exists.
  3. Separate the limit from the function valueIdentify the actual value independently of nearby behavior, including an undefined value.
  4. Recognize when a limit does not existDistinguish unequal approaches, unbounded behavior, and oscillation from a finite limit.

Topic 1.3Read limits from a graph

  1. Read a one-sided limit from a graphFollow the correct branch as the input approaches from the requested side.
  2. Decide whether both sides agreeCompare the graph's one-sided approaches to determine the two-sided limit.
  3. Read the value at the pointUse filled and open points correctly, separately from the limit.
  4. Recognize unbounded behavior on a graphIdentify the direction of an infinite limit near a vertical asymptote.

Topic 1.4Read limits from data

  1. Read the trend from each sideSort nearby input values by approach direction and estimate each limiting output.
  2. Compare left and right estimatesDecide whether the displayed trends suggest a common limit.
  3. Separate a reading from a limitDistinguish the value at the target from values sampled near it.
  4. State what the data supportsRecognize that finitely many readings alone do not prove an exact limit.

Topic 1.5Combine known limits

  1. Combine limits using arithmeticEvaluate sums, differences, and constant multiples from supplied limits.
  2. Use product and quotient lawsCalculate a product or quotient limit and check that a quotient's denominator limit is nonzero.
  3. Pass a limit through a continuous functionEvaluate a power, root, or composition while checking its domain and continuity.
  4. Recognize when a limit law is inconclusiveIdentify missing conditions without treating a failed rule as proof that the limit does not exist.

Topic 1.6Rewrite an indeterminate expression

  1. Recognize an indeterminate formDistinguish zero over zero from a limit that direct substitution settles.
  2. Factor and cancel to reveal a limitRemove a common factor only away from the approached input, then evaluate the limit.
  3. Use a conjugate with radicalsRationalize the expression and evaluate the resulting limit.
  4. Simplify a complex quotientCombine fractions or expand a difference quotient without changing nearby values.

Topic 1.7Choose a limit method

  1. Choose a useful first stepDecide whether substitution, factoring, rationalizing, or a standard limit fits the expression.
  2. Use standard trigonometric limitsAccount for angle scale factors and radian measure.
  3. Combine methods when necessarySimplify the expression before applying the appropriate limit rule.
  4. Check one-sided or domain restrictionsIdentify when the direction of approach changes the method or result.

Topic 1.8Trap a function between bounds

  1. Choose valid upper and lower boundsSelect inequalities that hold near the approached input, including on both sides when needed.
  2. Check that the bounds meetEvaluate the bounding limits and verify that they agree.
  3. Use the bounds to find the limitDetermine the trapped limit, including an oscillating function with shrinking amplitude.
  4. Recognize when the squeeze argument failsReject bounds that do not hold or do not approach the same value.

Topic 1.9Connect limit representations

  1. Match a formula to nearby graph behaviorConnect an algebraic expression with its approaches and domain restrictions.
  2. Connect a table, graph, and limit statementIdentify which representations describe the same local behavior.
  3. Locate a hole and its limiting valueGive the missing point's coordinates while keeping the actual function value separate.
  4. Choose a formula with the required behaviorSelect or construct a bounded expression that has the specified hole or limit.

Topic 1.10Classify breaks in a graph

  1. Locate a discontinuityIdentify inputs where continuity fails using values and one-sided limits.
  2. Recognize a removable holeCheck that a finite two-sided limit exists despite a missing or different point value.
  3. Recognize a jumpIdentify different finite one-sided limits.
  4. Recognize an infinite discontinuityIdentify unbounded behavior rather than a removable hole.
  5. Distinguish a corner from a breakRecognize that a connected graph can be continuous without being differentiable.

Topic 1.11Continuity at one point

  1. Check the three continuity conditionsCheck the point value, existence of the limit, and equality between them.
  2. Check a piecewise joinCompare the left limit, right limit, and assigned value at a joining point.
  3. Find a parameter that makes a join continuousSolve the matching-value condition and verify it at the point.
  4. Explain why continuity failsSelect the failed condition without requiring matching derivatives.

Topic 1.12Continuity on a domain

  1. Find where an expression is definedApply denominator, square-root, and logarithm restrictions correctly.
  2. Combine domain restrictionsIntersect the restrictions in products, quotients, and compositions.
  3. State intervals of continuityGive the intervals on which the function is continuous with correct endpoint notation.
  4. Check continuity at a domain endpointUse the available one-sided approach and include an endpoint only when appropriate.

Topic 1.13Fill a removable hole

  1. Decide whether you can fill a breakCheck that the two-sided limit exists and is finite.
  2. Find the missing valueSimplify the expression or use a known limit to determine the required output.
  3. Define a continuous extensionAssign the limiting value at the hole and preserve the rest of the function.
  4. Recognize a break that one value cannot repairReject a jump or infinite discontinuity as a removable hole.

Topic 1.14Infinite limits and vertical asymptotes

  1. Find candidate vertical asymptotesIdentify the zeros of the denominator or other boundary inputs. Then check the behavior near them.
  2. Determine the sign on each sideUse a sign analysis to distinguish positive and negative infinite limits.
  3. Distinguish a hole from an asymptoteCheck what remains after cancellation rather than treating every denominator zero as a pole.
  4. Connect limits with asymptote behaviorMatch the one-sided limit statements to the graph near the asymptote.

Topic 1.15Behavior far from the origin

  1. Find the end behavior of a rational functionCompare degrees and leading coefficients.
  2. Handle positive and negative infinity separatelyTreat radicals and absolute values correctly on each tail.
  3. Compare common growth ratesDetermine which terms dominate in polynomial, exponential, and logarithmic expressions.
  4. Connect end behavior with asymptotesState a horizontal asymptote or identify the absence of a finite horizontal limit.

Topic 1.16Guarantee an intermediate value

  1. Check the Intermediate Value Theorem conditionsVerify continuity throughout the closed interval and compute the endpoint values.
  2. Identify a guaranteed intermediate valueCheck that the target lies between the endpoint outputs.
  3. Choose the strongest justified conclusionState existence in the appropriate interval without claiming uniqueness.
  4. Recognize when the theorem cannot decideDistinguish a failed hypothesis from proof that no solution exists.